# Critical slowing down in the Sherrington--Kirkpatrick model

*An approachable but complete proof.*

---

## 1. The model

Let n be a large positive integer. A spin configuration is a point


x=(x_1,...,x_n)in{-1,+1}^n .


The Sherrington--Kirkpatrick (SK) interaction is a random symmetric matrix W=(W_{ij})_{1le i,jle n} with


W_{ij}sim N(0,1/n)(i<j),

W_{ii}sim N(0,2/n),

W_{ij}=W_{ji}.


Thus W is a Gaussian orthogonal ensemble (GOE) matrix. The diagonal entries do not affect the spin model, because x_i^2=1 for every spin. They are included only because they make the matrix spectrum behave exactly like GOE; we will remove them from the Gibbs law whenever convenient.

The Hamiltonian is


H_W(x)=frac12 x^T W x
=sum_{i<j}W_{ij}x_ix_j+frac12sum_i W_{ii}.


The last term is the same for every configuration x, so it cancels from every probability ratio. The Gibbs measure is


mu_W(x)=frac{e^{H_W(x)}}{
sum_{yin{-1,+1}^n}e^{H_W(y)}
}
propto
exp!(sum_{i<j}W_{ij}x_ix_j).


We often write mu=mu_W when the disorder W is fixed.

The number beta is the inverse temperature. In this paper the inverse temperature is exactly beta=1, the critical value. For beta<1 the model is in the high-temperature phase and mixes quickly; at beta=1 the spectral edge is critical and slow modes appear on scale n^{-2/3}.

---

## 2. Heat-bath dynamics

A heat-bath update at site i replaces x_i by a fresh spin chosen from its conditional law given all other spins. Since a spin has only two possible values, this means:


x_i=
+1 & with probability dfrac{e^{g_i(x)}}{2cosh g_i(x)},[2mm]
-1 & with probability dfrac{e^{-g_i(x)}}{2cosh g_i(x)},


g_i(x)=sum_{jne i}W_{ij}x_j .


Equivalently, the new spin has conditional mean tanh g_i(x).

We use two clocks.

* **Continuous time.** Each site has an independent rate-one exponential clock. When its clock rings, that site is refreshed. The generator is


mathcal L=calL=sum_{i=1}^n(P_i-Id),

P_if(x)=mathbb E_bbE_mu[f(x)mid x_{-i}],


where x_{-i} means all spins except x_i.

* **Discrete time.** At each step one site is chosen uniformly at random and refreshed. The transition operator is


P=frac1nsum_{i=1}^n P_i .


These clocks are related by


mathcal L=n(P-Id).calL=n(P-Id).


So one unit of continuous time corresponds roughly to n attempted discrete updates, but the exact relation is the displayed identity.

For a starting configuration x, let


d_t^{ct}(W,x)
=|e^{tmathcal L}calL(x,cdot)-mu_W|_{TV},

d_k^{dt}(W,x)
=|P^k(x,cdot)-mu_W|_{TV},


where total variation is


|nu-mu|_{TV}
=frac12sum_x|nu(x)-mu(x)|
=sup_{A⊆{-1,+1}^n}|nu(A)-mu(A)|.


The worst-start mixing times are


t_{mix}
=inf{tge0:max_x d_t^{ct}(W,x)le1/4},



k_{mix}
=min{kinmathbb NbbNcup{0}:max_x d_k^{dt}(W,x)le1/4}.


All probabilistic statements below are over the random matrix W (and, when we temporarily add an auxiliary Gaussian diagonal, over that auxiliary randomness as well).

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## 3. The three main results

### Theorem A (typical equilibrium starts are slow)

For every deterministic sequence t_nge0 with


t_n=o(n^{2/3}),


we have


mu_W{x:d_{t_n}^{ct}(W,x)>1/4}
→ 1
qquadin probability over W.


For every deterministic integer sequence k_nge0 with


k_n=o(n^{5/3}),


we have


mu_W{x:d_{k_n}^{dt}(W,x)>1/4}
→ 1
qquadin probability over W.


The theorem tests each realized equilibrium starting configuration separately. If instead one averaged the transition law over the initial configuration drawn from mu_W, stationarity would make the law equal to mu_W at every time. The theorem says that this averaging hides a split: most fixed starts are still far away.

### Theorem B (covariance and linear tests)

Let


Σ=Cov_{mu_W}(x)=mu_W[xx^T],


and let


mathcal D_calD_{mu_W}(f)
=sum_{i=1}^nmu_W[Var(fmid x_{-i})]


be the unscaled Dirichlet form. For linear functions a^Tx, define


mathcal R_calR_{lin}(mu_W)
=sup_{ainmathbb R^n∖bbR^n∖{0}}
frac{Var_{mu_W}(a^Tx)}
{mathcal D_calD_{mu_W}(a^Tx)} .


For every positive deterministic sequence M_nto∞, with probability tending to one both |Σ| and mathcal R_calR_{lin}(mu_W) lie in


[
frac{n^{2/3}}{M_n},
 M_n n^{2/3}
].


The factor M_n can diverge as slowly as we like. The theorem only controls linear functions; it does not by itself control every function.

### Theorem C (critical mixing bounds)

For every fixed ε>0,


mathbb PbbP{
n^{2/3-ε}le t_{mix}le e^{ε n}
}→ 1,



mathbb PbbP{
n^{5/3-ε}le k_{mix}le e^{ε n}
}→ 1.


The lower bounds come from Theorem A. The upper bounds are proved by comparison with one fixed high-temperature model and are exponentially weak. The sharper matching exponents are the subject of the companion paper, but the present paper proves the displayed bounds completely.

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## 5. Elementary identities

Before proving anything hard, we record the bookkeeping that makes every later step work.

### Lemma 5.1 (Heat-bath forms)

For every function f:{-1,+1}^ntomathbb R,bbR, each P_i is an orthogonal projection in L^2(mu). The unscaled Dirichlet form is


mathcal D_calD_mu(f)
=-< f,mathcal Lf>_calLf>_mu
=sum_{i=1}^nmu[(f-P_if)^2]
=sum_{i=1}^ninf_{h_i}mu[(f-h_i(x_{-i}))^2],


where the infimum is over all functions h_i of x_{-i}. The discrete form is


< f,(Id-P)f>_mu=frac1nmathcal D_calD_mu(f),


and the spectrum of P lies in [0,1].

For a linear function a^Tx,


mathcal D_calD_mu(a^Tx)=sum_{i=1}^n a_i^2alpha_i,

alpha_i=mu[sech^2 g_i],

g_i(x)=sum_{jne i}W_{ij}x_j .


#### Proof

Conditional expectation onto functions of x_{-i} is the orthogonal projection in L^2(mu). Therefore


< f,(Id-P_i)f>_mu
=|f-P_if|_{L^2(mu)}^2,


and this squared distance is the infimum over all functions of x_{-i}.

Summing over i gives the formula for mathcal D_calD_mu. Averaging instead gives


< f,(Id-P)f>_mu
=frac1nsum_i|f-P_if|_{L^2(mu)}^2
=frac1nmathcal D_calD_mu(f).


Since each P_i is an orthogonal projection, 0le< f,P_if>_mule< f,f>_mu. Averaging over i gives the same inequality for P, so the spectrum of P lies in [0,1].

For a linear function, everything except a_ix_i is fixed once x_{-i} is fixed. Under mu(cdotmid x_{-i}), the spin x_i has mean tanh g_i and variance


1-tanh^2 g_i=sech^2 g_i .


Thus the contribution of site i to mathcal D_calD_mu(a^Tx) is


a_i^2mu[Var(a_ix_imid x_{-i})]
=a_i^2mu[sech^2 g_i].


Summing proves the formula. QED

---

### Lemma 5.2 (Stationary increments)

Fix W and a function f:{-1,+1}^ntomathbb R.bbR. Let (X_t)_{tge0} be the continuous-time chain and (Y_k)_{kge0} the discrete-time chain, each started from mu. Then


mathbb E_bbE_mu[(f(X_t)-f(X_0))^2]
le2t,mathcal D_calD_mu(f),
 tge0,


and


mathbb E_bbE_mu[(f(Y_k)-f(Y_0))^2]
le(2k)/(n)mathcal D_calD_mu(f),
 k=0,1,2,... .


In particular, for g(x)=v^Tx with |v|=1, the right-hand sides are at most 2t and 2k/n.

#### Proof

Stationarity gives


mathbb E_bbE_mu[(f(X_t)-f(X_0))^2]
=2< f,(Id-e^{tmathcal L}calL)f>_mu .


The operator -mathcal LcalL is self-adjoint and nonnegative. Diagonalize it in an orthonormal eigenbasis with eigenvalues lambdage0. The multiplier of the f-component in eigenvalue lambda is


1-e^{-tlambda}le tlambda .


Therefore


2< f,(Id-e^{tmathcal L}calL)f>_mu
le2t< f,-mathcal Lf>_calLf>_mu
=2tmathcal D_calD_mu(f).


For the discrete chain, P is self-adjoint with spectrum in [0,1]. The stationary increment is


2< f,(Id-P^k)f>_mu .


For 0lethetale1,


1-theta^kle k(1-theta).


Also


< f,(Id-P)f>_mu=frac1nmathcal D_calD_mu(f).


This proves the discrete bound. Finally, Lemma 5.1 gives


mathcal D_calD_mu(v^Tx)
=sum_i v_i^2mu[sech^2 g_i]
lesum_i v_i^2=1.


QED

---

### Lemma 5.3 (Overlap controls covariance)

Zero field gives mu(x)=mu(-x), so every spin has mean zero. Hence


Σ=Cov_mu(x)=mu[xx^T],

Σ_{ii}=1.


Let


q(x,y)=(x^Ty)/(n)


be the overlap of two configurations. For independent x,ysimmu,


Tr(Σ^2)
=n^2mathbb E_bbE_{mu ⊗ mu}[q(x,y)^2].


#### Proof

Expand the square:


Tr(Σ^2)
=sum_{i,j}Σ_{ij}^2
=sum_{i,j}mu[x_ix_j]^2 .


For independent x,ysimmu,


mathbb E_bbE_{mu ⊗ mu}[q(x,y)^2]
=frac1{n^2}sum_{i,j}mu[x_ix_j]^2 .


Multiplying by n^2 proves the identity. QED


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## 6. Spectral input near the critical edge

This section proves the matrix estimates that feed the spherical model. The reader should keep one picture in mind: GOE eigenvalues near the edge 2 are spaced like n^{-2/3}. We will look a tiny distance


s_0=L_n n^{-2/3}


above the edge, where L_nto∞ very slowly. The slowly growing factor L_n is essential: it gives enough room to prove estimates without destroying the sharp n^{2/3} scale.

### 6.1 The scales

Let L_nto∞ be deterministic with


L_nlelog n


for all sufficiently large n. Define


s_*=n^{-2/3},

s_0=L_n s_*=L_n n^{-2/3},



h_n=L_n^{1/8},

A_n=h_n+√(log(2+L_n)),

eta_n=A_n L_n^{-3/4}.


Then eta_nto0. Define the deterministic reference function


R(s)=frac{2+s-√(s(4+s))}2,
 s>0.


Note that R(s)=1-sqrt s+O(s) as s↓0.

### Proposition 6.1 (An eigenvalue-only good event)

Let W be the GOE matrix of Section 1, with eigenvalues


lambda_1gelambda_2ge...gelambda_n .


There are events mathcal G_n(L_n),calG_n(L_n), measurable with respect to the eigenvalues alone, such that


mathbb P(bbP(mathcal G_n(L_n)^c)calG_n(L_n)^c)le Cexp(-cL_n^{1/4})


for all sufficiently large n. On mathcal G_n(L_n),calG_n(L_n), for every fixed M<∞,


lambda_j=2+o(s_0)(1le jle3),

max_i|lambda_i|le3,


and, for s_0le sle M,


frac1nsum_{i=1}^nfrac1{2+s-lambda_i}
=R(s)+o(sqrt s),



frac1nsum_{i=1}^nfrac1{(2+s-lambda_i)^2}
le(C_M)/(sqrt s).


All denominators are positive. The little-oh errors are deterministic on the event: the first three eigenvalue errors divided by s_0 are at most


C(L_n^{-1}+eta_n),


and the first resolvent error divided by sqrt s is at most


C_M(L_n^{-3/2}+eta_n).


The event does not depend on M.

The rest of Section 6 proves this proposition.

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### 6.2 Tridiagonal reduction

The first step is classical: a GOE matrix has the same eigenvalues as a random tridiagonal matrix. Tridiagonal matrices are much easier because their entries are independent and because their resolvents satisfy a three-term recurrence.

#### Lemma 6.2 (Tridiagonal representation)

The eigenvalues of W have the same law as the eigenvalues of a symmetric tridiagonal matrix T with independent entries


T_{ii}sim N(0,2/n),

T_{i,i+1}=T_{i+1,i}simfrac{chi_{n-i}}{sqrt n}
(1le i<n),


where chi_m is the Euclidean norm of m independent standard normal variables.

#### Proof

Expose one column of W at a time.

The first diagonal entry, the off-diagonal entries in the first column, and the trailing (n-1)times(n-1) block are independent. The first column has n-1 independent N(0,1/n) off-diagonal entries. Choose an orthogonal transformation of the trailing n-1 coordinates that sends this column vector to its nonnegative length times the first coordinate vector. Conjugating by this transformation preserves eigenvalues and produces the desired first off-diagonal entry, whose law is chi_{n-1}/sqrt n.

The trailing block has density proportional to exp(-n,Tr(B^2)/4). This density and Euclidean volume are invariant under orthogonal conjugation. Conditional on the column vector, the chosen transformation is fixed, so the conjugated trailing block is still GOE and is independent of the entries already exposed.

Repeat the construction inside the trailing block, then inside each successive block. At step i, the fresh column has n-i independent N(0,1/n) entries. This gives the stated distributions and their joint independence. QED

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### 6.3 Concentration of chi entries

We need two estimates for centered chi variables. For mge1, let chi_m be as above. Then for every real t,


mathbb EbbEexp(t(chi_m-mathbb EbbEchi_m)/(sqrt n))
leexp(Ct^2/n),


and


0lesqrt m-mathbb EbbEchi_mle(C)/(sqrt m).


#### Proof

The Gaussian-square moment-generating function and Chernoff's inequality give


mathbb PbbP{chi_m^2ge ma}le(ae^{1-a})^{m/2}
(a>1),


and the same right-hand side for the lower-tail event when 0<a<1. Since


a-1-log age(sqrt a-1)^2,


we obtain


mathbb PbbP{|chi_m-sqrt m|ge r}le2e^{-r^2/2}
(rge0).


This tail bound gives Var(chi_m)le C. By Jensen's inequality,


sqrt m-mathbb EbbEchi_m
=(Var(chi_m))/(sqrt m+mathbb EbbEchi_m)
le(C)/(sqrt m).


For the exponential bound, let X,X' be independent copies of chi_m and put Z=X-X'. Then Z is symmetric and


mathbb PbbP{|Z|ge r}le4e^{-r^2/8},

mathbb EbbE Z^{2k}le4cdot8^k k!
(kge1).


Jensen's inequality conditional on X, followed by the power series of cosh, gives


mathbb EbbE e^{t(X-mathbb EbbE X)}
lemathbb EbbE e^{tZ}
=mathbb Ecosh(tZ)bbEcosh(tZ)
le1+4sum_{kge1}frac{8^k k!t^{2k}}{(2k)!}
le e^{32t^2}.


Replacing t by t/sqrt n proves the exponential bound. The centered diagonal entries satisfy the same estimate directly. QED


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### 6.4 The noise is controlled by a deterministic energy

We now compare the random tridiagonal matrix T with a deterministic tridiagonal matrix D whose off-diagonal entries are the means of the random ones.

Let D be the deterministic symmetric tridiagonal matrix with zero diagonal and


D_{i,i+1}=D_{i+1,i}=c_i:=√(1-i/n)
(1le i<n).


Set


E=2I-D.


Then E is tridiagonal, has diagonal entries 2, and has off-diagonal entries -c_i. It is positive definite, and 0le Ele4I.

The random matrix is T from Lemma 6.2. Put


H=T-D.


The entries of H are centered: the diagonal entries are centered Gaussian, and the off-diagonal entries are


frac{chi_{n-i}}{sqrt n}-c_i .


#### Lemma 6.4 (Noise controlled by deterministic energy)

There is a universal constant C_0 such that, with


Phi_n(x)
=C_0n^{-1/2}A_n(x+s_0)^{1/4}
√(1+log(1+x/s_0)),
 xge0,


the following holds. On an event mathcal F_ncalF_n with


mathbb P(bbP(mathcal F_n^c)calF_n^c)le Cexp(-cL_n^{1/4}),


we have, simultaneously for every unit vector uinmathbb R^n,bbR^n,


|u^THu|lePhi_n(u^TEu).


Moreover,


sup_{xge0}|Phi_n'(x)|le Ceta_n,

Phi_n(0)le Ceta_ns_0.


#### Proof

The proof has three steps.

1. The deterministic energy u^TEu controls where a unit vector can concentrate and how fast it can oscillate.
2. One high-probability event controls block prefixes of the random entries at every relevant scale.
3. Summation by parts turns those prefix bounds into a quadratic-form bound.

##### Step 1: energy controls mass and gradients

For a unit vector u, write


e=u^TEu.


Expanding the quadratic form gives


e=sum_{i<n}c_i(u_i-u_{i+1})^2
+sum_{i=1}^n(2-c_i-c_{i-1})u_i^2,


where we set c_0=c_n=0.

The row sums of D are at most 2, so 0le Ele4I. The coefficients 2-c_i-c_{i-1} are at least i/(2n): for interior indices this follows from 1-√(1-t)ge t/2, and the boundary indices satisfy the same inequality directly. Also c_ige1/sqrt2 when ile n/2.

Therefore


sum_i(i)/(n)u_i^2le2e,


and


sum_{i>n/2}u_i^2+sum_{i<n}(u_i-u_{i+1})^2le Ce.


For the second inequality, control gradients in the first half by the weighted gradient term, and use


(u_i-u_{i+1})^2le2u_i^2+2u_{i+1}^2


for the remaining edges.

##### Step 2: one event for every energy scale

The vector u will be chosen after the matrix is realized, so the probability estimate must be simultaneous over all energy scales and all spatial shells.

Set


sigma_j=2^js_*,

0le jle J_n:=leftlceillog_2(4/s_*)rightrceil,


and


d_j=max{1,lfloorsigma_j^{-1/2}rfloor}.


For each of the two independent entry sequences (diagonal entries and centered off-diagonal entries), partition its index set into consecutive blocks of lengths between d_j and 2d_j. Such a partition is obtained by merging a last short block with its predecessor.

Assign label r=0 when the first index of a block is at most 2nsigma_j. Otherwise assign label rge1 when the first index lies in


(2^rnsigma_j,2^{r+1}nsigma_j].


If xi_i is either centered entry sequence and a block has length mle2d_j, its partial sums


S_k=xi_1+...+xi_k


obey


mathbb PbbP{max_{kle m}|S_k|ge a}
le2exp(-(na^2)/(Cd_j)).


Indeed, the exponential bound of Lemma 6.3 makes


exp(tS_k-(Ckt^2)/(n))


a nonnegative supermartingale. Stopping it at its first crossing of a bounds the one-sided probability by


exp(-ta+(Cmt^2)/(n)).


Optimizing t>0 and repeating for -S_k proves the displayed estimate.

At a fixed pair (j,r), the number of blocks is at most


C(1+(2^rnsigma_j)/(d_j))
le C2^rnsigma_j^{3/2}
=C2^{r+3j/2}.


For sigma_jle1, use d_jgesigma_j^{-1/2}/2. For sigma_j>1, use d_j=1. The additive constant is absorbed because nsigma_j^{3/2}ge1.

Consequently, for a sufficiently large fixed constant K, there is an event mathcal F_ncalF_n on which every block of both sequences has all its centered partial sums bounded by


K√(d_j/n),(h_n+√(j+r)).


Moreover,


mathbb P(bbP(mathcal F_n^c)calF_n^c)
le Csum_{j,rge0}2^{r+3j/2}
e^{-cK^2(h_n+√(j+r))^2}
le Ce^{-c'h_n^2}
=Ce^{-c'L_n^{1/4}}.


Here K is chosen so that the factors in j and r form summable geometric series. The bound does not acquire a factor from the number of scales.

##### Step 3: from block prefixes to quadratic forms

We now fix the noise event mathcal F_ncalF_n and choose an arbitrary unit vector u.

Select the first j for which


elesigma_j.


Since ele4, this index exists, and


sigma_jle2(e+s_*).


For the weights w_i=u_i^2 or w_i=u_iu_{i+1}, the sum of their absolute values over all blocks with label r is at most C2^{-r}.

For rge1, every coordinate used is beyond 2^rnsigma_j. The energy-tail bound gives total squared mass at most


(2e)/(2^rsigma_j)le2^{1-r}.


The off-diagonal weights obey the same bound because


|u_iu_{i+1}|lefrac{u_i^2+u_{i+1}^2}{2}.


For r=0, use the unit norm.

The summed variation of these weights within those blocks is at most


C2^{-r/2}sqrt e.


To verify this, use


u_{i+1}^2-u_i^2=(u_{i+1}-u_i)(u_{i+1}+u_i)


for diagonal weights and


u_{i+1}u_{i+2}-u_iu_{i+1}
=u_{i+1}[(u_{i+2}-u_{i+1})+(u_{i+1}-u_i)]


for off-diagonal weights. Cauchy--Schwarz combines the tail mass bound with the total gradient bound. Each coordinate or edge is counted at most a fixed number of times.

For clarity, the summation-by-parts step can be made blockwise. Suppose all centered entry prefixes in a block are bounded by B, and let barxi be the entry average on that block. Then


|barxi|le B/d_j.


The prefixes of xi_i-barxi are at most 2B and vanish at the block end. Therefore


|sum_{i in block}xi_iw_i|
le(B)/(d_j)sum_{i in block}|w_i|
+2Bsum_{substack{i,i+1 in block}}|w_{i+1}-w_i|.


Using the prefix bound and summing by labels proves that either centered weighted sum is bounded by


Cn^{-1/2}sum_{rge0}(h_n+√(j+r))
(d_j^{-1/2}2^{-r}+d_j^{1/2}2^{-r/2}sqrt e)
le C(h_n+sqrt j)n^{-1/2}sigma_j^{1/4}.


For sigma_jle1, the last step uses


d_jasympsigma_j^{-1/2},

elesigma_j.


For 1<sigma_j<8, it follows from d_j=1 with a changed universal constant. The series in r converges geometrically, including its sqrt r factor.

The centered entries have now been controlled uniformly in u. The only remaining part of T-D is the deterministic error in the off-diagonal means. By Lemma 6.3, their difference from c_i is at most


frac{C}{sqrt n,√(n-i)}.


It is at most C/n for ile n/2 and at most C/sqrt n thereafter. Thus their quadratic-form contribution is at most


C/n+Ce/sqrt n,


using again the tail mass bound.

Finally,


jle Clog(2+e/s_*)
le C[log(2+L_n)+log(1+e/s_0)].


Together with sigma_jle2(e+s_*) and s_*le s_0, this converts the centered estimate into


|u^THu|lePhi_n(e).


The mean errors are absorbed into the same bound because 0le ele4 and L_nge1 for large n. Both entry sequences, and the factor two from off-diagonal entries, are absorbed by choosing C_0 once and for all.

##### Step 4: smallness of the comparison function

Finally, put


q(x)=1+log(1+x/s_0).


Direct differentiation gives


Phi_n'(x)
=C_0n^{-1/2}A_n(x+s_0)^{-3/4}
(frac{√(q(x))}4+frac1{2√(q(x))}).


Since y^{-3/4}√(1+log y) is bounded for yge1, its supremum is at most


Cn^{-1/2}A_ns_0^{-3/4}
=Ceta_n.


Also


Phi_n(0)/s_0=C_0eta_n.


This proves the two scalar bounds. QED


---

### 6.5 The deterministic resolvent

Lemma 6.4 reduces the random trace problem to the deterministic matrix D. Its entries were chosen so that its characteristic polynomials satisfy a Hermite recurrence. That recurrence lets us compute its resolvent very accurately near the edge.

#### Lemma 6.5 (Hermite resolvent estimates)

Let


e_1le e_2le...le e_n


be the eigenvalues of


E=2I-D.


Set


R_D(s)=frac1nTr(s+E)^{-1},

T_D(s)=frac1nTr(s+E)^{-2}
=-R_D'(s).


For every fixed M<∞ and all sufficiently large n, uniformly for


s_0le sle M,


we have


T_D(s)le C_Ms^{-1/2},

0le R(s)-R_D(s)le(C_M)/(ns).


Moreover,


N(t):=#{i:e_ile t}le Cnt^{3/2}
(tge s_0).


#### Proof

Reverse the coordinates of D. Its off-diagonal entries become


√(k/n),

1le k<n.


Let P_k(z) be the characteristic polynomial of the ktimes k principal minor obtained from the first k reversed coordinates. Then


P_0(z)=1,

P_1(z)=z,

P_{k+1}(z)=zP_k(z)-(k)/(n)P_{k-1}(z).


Induction gives


P_k'=kP_{k-1}.


Differentiating the recurrence and using the recurrence one step earlier shows that


P_{k+1}'=P_k+kP_k.


Consequently,


P_n''-nzP_n'+n^2P_n=0.


Since P_n is the characteristic polynomial of D,


R_D(s)=(P_n'(2+s))/(nP_n(2+s)).


The differential equation therefore yields


R_D(s)^2-(2+s)R_D(s)+1=(T_D(s))/(n)>0.


For every s>0, R_D(s) is finite because Ege0.

The two roots of the quadratic on the left are


R(s)=frac{2+s-√(s(4+s))}2


and


R_+(s)=frac{2+s+√(s(4+s))}2.


At large s, R_D(s)to0 whereas R_+(s)to∞. Positivity in the Riccati equation thus places R_D below the smaller root there, and continuity prevents it from crossing either root anywhere on (0,∞).

If


delta(s)=R(s)-R_D(s),


then factoring the quadratic gives


0ledelta(s)le
frac{T_D(s)}{n√(s(4+s))}.


Termwise comparison of resolvents shows that


T_D(s)lefrac2s[R_D(s/2)-R_D(s)]
lefrac2s[R(s/2)-R(s)+delta(s)].


Differentiating the explicit expression for R gives


0<-R'(s)
=frac{2}{√(s(4+s)),
(2+s+√(s(4+s)))}
lefrac1{2sqrt s}.


Integrating from s/2 to s therefore gives


R(s/2)-R(s)lesqrt s-√(s/2)lesqrt s.


Hence


T_D(s)le(C)/(sqrt s)
+frac{2T_D(s)}{ns√(s(4+s))}.


For sge s_0, the coefficient of the last term is at most


(ns_0^{3/2})^{-1}=L_n^{-3/2}to0.


Absorbing it proves the bound on T_D. The displayed bound on delta then follows from the root-difference inequality. Notice that R_D(s/2)le R(s/2) was established for every positive s/2; it does not require s/2ge s_0.

The second-trace estimate also gives the counting estimate needed for the later dyadic transfer.

For s_0le tle1, each e_ile t contributes at least (2t)^{-2} to nT_D(t). Thus


N(t)le4nt^2T_D(t)le Cnt^{3/2}.


For tge1, the same bound follows from N(t)le n. QED


---

### 6.6 Eigenvalue comparison and proof of Proposition 6.1

We now transfer the quadratic-form bound of Lemma 6.4 and the deterministic resolvent estimates of Lemma 6.5 to the actual eigenvalues of W.

#### Step 1: ordered eigenvalues

Work first in the tridiagonal realization T.

On the event mathcal F_ncalF_n from Lemma 6.4, the form bound controls the Rayleigh quotient of


E-H=2I-T


between


f_-(e):=e-Phi_n(e)
qquadandqquad
f_+(e):=e+Phi_n(e).


By sup|Phi_n'|le Ceta_n and eta_nto0, both functions are increasing for large n.

The span of the first i eigenvectors of E has every unit-vector energy at most e_i. The min--max principle therefore gives


2-lambda_ile f_+(e_i).


On the span of eigenvectors i through n, every energy is at least e_i. The complementary max--min principle gives


2-lambda_ige f_-(e_i).


Therefore


|2-lambda_i-e_i|lePhi_n(e_i),
 1le ile n.


Define mathcal G_n(L_n)calG_n(L_n) to be precisely the event on which this inequality holds, with the deterministic e_i and Phi_n constructed above. It depends only on the ordered eigenvalues. The tridiagonal event mathcal F_ncalF_n implies it, so Lemma 6.2 transfers the probability bound to W. All remaining conclusions are deterministic on mathcal G_n(L_n).calG_n(L_n).

#### Step 2: the first three eigenvalues

The conditioning argument will need three large coordinate variances. To locate the corresponding eigenvalues, take


m=lfloor n^{1/3}rfloor


and the span of the first three discrete Dirichlet sine modes on {1,...,m}, extended by zero.

The Dirichlet Laplacian on this segment has eigenvalues


4sin^2((kpi)/(2(m+1))),


so its quadratic form on this span is at most C/m^2. On vectors supported on the segment, the form difference between E and that Laplacian has absolute value at most


2max_{1le i<m}|1-c_i|sum_{i<m}|u_iu_{i+1}|
le Cm/n.


The min--max principle therefore gives


0le e_1,e_2,e_3le C(m^{-2}+m/n)le Cn^{-2/3}.


The comparison inequalities and Phi_n(0)le Ceta_ns_0 imply


max_{jle3}(|lambda_j-2|)/(s_0)
le C(L_n^{-1}+eta_n).


Also every e_iin[0,4], so


max_i|lambda_i|le2+Phi_n(4).


The assumption L_nlelog n gives Phi_n(4)=o(1), proving the stated bound by 3.

#### Step 3: the resolvent traces

The eigenvalue comparison also controls every denominator: for every sge s_0 and every ein[0,4],


(Phi_n(e))/(s+e)
le(Phi_n(0)+Ceta_ne)/(s+e)
le Ceta_n.


Thus


2+s-lambda_i=(s+e_i)(1+O(eta_n)),


uniformly in both i and sge s_0.

The normalized second trace is consequently at most a constant times T_D(s), proving its bound in Proposition 6.1.

For the first trace, subtract term by term and use


(e_i+s_0)^{1/4}le(s+e_i)^{1/4}


to obtain


|frac1nsum_ifrac1{2+s-lambda_i}-R_D(s)|
le Cn^{-1/2}A_nfrac1nsum_i
(s+e_i)^{-7/4}√(1+log(1+e_i/s_0)).


We spell out the dyadic sum because its uniformity near s_0 is useful.

The range e_ile s contributes at most


Cs^{-7/4}frac{N(s)}n√(1+log(1+s/s_0))
le Cs^{-1/4}√(1+log(1+s/s_0)).


For kge0, the shell


2^ks<e_ile2^{k+1}s


contributes at most


Cs^{-1/4}2^{-k/4}
√(1+log(1+s/s_0)+(k+1)log2),


by the counting estimate of Lemma 6.5. That counting estimate remains valid above one, so all shells are covered.

The convergent geometric series, with its additional square-root factor, bounds the sum by


C_Ms^{-1/4}√(1+log(1+s/s_0)).


Dividing the resulting trace error by sqrt s and writing r=s/s_0 gives


C_Meta_nr^{-3/4}√(1+log(1+r))
le C_Meta_n
(rge1).


Finally, Lemma 6.5 gives


(|R_D(s)-R(s)|)/(sqrt s)
lefrac{C_M}{ns^{3/2}}
le C_ML_n^{-3/2}.


This proves the quantitative first-trace error and completes all of Proposition 6.1. The event was defined before M was chosen, and each displayed error is deterministic on that event. Consequently every deterministic sequence of spectra in mathcal G_n(L_n)calG_n(L_n) satisfies the estimates for every fixed finite M, without any condition on eigenvector orientation. QED


---

## 7. The spherical model

The Ising cube is hard because its coordinates are discrete. The spherical model replaces the cube by a sphere of radius sqrt n, where Gaussian conditioning gives exact formulas. We then transfer spherical estimates back to the cube in Section 8.

### 7.1 Tilting the sphere

Fix a deterministic ordered spectrum


lambda_1gelambda_2ge...gelambda_n,
 nge3,
 max_i|lambda_i|le3,


and put


Λ=diag(lambda_1,...,lambda_n).


Let sigma_n be uniform probability measure on


{uinmathbb R^n:bbR^n:|u|^2=n}.


For 0le ale2, define


S(a)=int e^{a u^TΛ u/2},sigma_n(du),


and


B(a)=inf_{z>alambda_1}
{
(z-1)/(2)
-frac1{2n}sum_{i=1}^nlog(z-alambda_i)
}.


The associated tilted probability is


sigma_{Λ,a}(du)
=S(a)^{-1}e^{a u^TΛ u/2},sigma_n(du).


At a=1 we write


sigma_Λ=sigma_{Λ,1}.


Quadratic exponential laws on a sphere are Bingham distributions. The key fact is that they can be represented by independent Gaussian coordinates conditioned on their squared radius.

#### Lemma 7.1 (Gaussian representation)

For every ain[0,2], the infimum defining B(a) has a unique minimizer z_a. With


k_i(a)=(z_a-alambda_i)^{-1},

sum_{i=1}^n k_i(a)=n,


one has


1+alambda_nle z_ale1+alambda_1,

frac1{13}le k_i(a)le n.


Let Y_i be independent N(0,k_i(a)) variables, and let p_a be the density of


sum_{i=1}^n Y_i^2 .


Then


S(a)=e^{nB(a)}
frac{p_a(n)}{p_{chi_n^2}(n)},


where p_{chi_n^2} is the density of a chi-square variable with n degrees of freedom.

The conditional law of Y=(Y_1,...,Y_n) given |Y|^2=n, defined by radial disintegration, is sigma_{Λ,a}.

#### Proof

Denote the objective in the definition of B(a) by F_a(z). On z>alambda_1,


F_a'(z)=frac12-frac1{2n}sum_i(z-alambda_i)^{-1},


and


F_a''(z)=frac1{2n}sum_i(z-alambda_i)^{-2}>0.


The derivative tends to -∞ at the left endpoint and to 1/2 at infinity. The function itself tends to +∞ at both ends. This proves existence, uniqueness, and the saddle equation


sum_i(z_a-alambda_i)^{-1}=n.


The average of the reciprocals of z_a-alambda_i is one. Their minimum is therefore at most one, and their maximum is at least one. This gives the stated bounds on z_a. Consequently,


0<z_a-alambda_i
le1+a(lambda_1-lambda_n)le13.


The lower bound for k_i(a) follows. The upper bound follows from their positivity and sum n. At a=0, these formulas give z_0=1, k_i(0)=1, and B(0)=0.

Now fix the minimizer z=z_a. Put


Q=zI-aΛ,


and let Y^{z,a}sim N(0,Q^{-1}). Its density relative to a standard Gaussian vector at yinmathbb R^nbbR^n is


(det Q)^{1/2}
exp(-(z-1)/(2)|y|^2+frac a2y^TΛ y).


The standard Gaussian has uniform conditional angular law. Integrating this ratio over the sphere of squared radius n gives


frac{p_{z,a}(n)}{p_{chi_n^2}(n)}
=(det Q)^{1/2}e^{-n(z-1)/2}S(a),


where p_{z,a} is the density of |Y^{z,a}|^2.

This is a polar-coordinate identity between continuous radial densities at a positive radius. At z=z_a, its rearrangement is


S(a)=e^{nB(a)}
frac{p_a(n)}{p_{chi_n^2}(n)}.


The same density ratio, with the radius fixed, proves that the conditional law is sigma_{Λ,a}. QED

---

### 7.2 Two-square smoothing

We will repeatedly use a simple density bound. If G,H are independent standard Gaussians and d,e>0, then the density r_{d,e} of


dG^2+eH^2


satisfies


r_{d,e}(t)
=frac1{2pi√(de)}
int_0^t
frac{e^{-x/(2d)-(t-x)/(2e)}}{√(x(t-x))},dx
lefrac1{2√(de)},
 t>0.


The integral of [x(t-x)]^{-1/2} over (0,t) is pi. Convolving with further independent random variables preserves this upper bound.

---

### 7.3 Density at the mean

The next estimate is needed at the mean of a weighted sum of squares, including arrays with a dominant weight.

#### Lemma 7.2 (Density at the mean)

There is a universal constant c>0 such that, for every nge3 and every k_1,...,k_n>0 with


sum_{i=1}^n k_i=n,


the density p of


sum_{i=1}^n k_iG_i^2,


for independent standard Gaussians G_i, satisfies


p(n)ge c(sum_{i=1}^n k_i^2)^{-1/2}.


#### Proof

After normalization, it is enough to exclude a sequence of densities vanishing at zero. There are two ways such a sequence of weights can behave: all weights vanish individually, giving a Gaussian limit, or a positive largest weight remains and supplies a density bounded below near zero.

Set


tau=(sum_i k_i^2)^{1/2},

d_i=k_i/tau .


The centered random variable


T_d=sum_i d_i(G_i^2-1)
=(sum_i k_iG_i^2-n)/(tau)


has density f_d with


f_d(0)=tau p(n),

sum_i d_i^2=1.


Thus it suffices to bound f_d(0) below uniformly over all such finite arrays. Each array has at least three positive weights. Its characteristic function is integrable, since its absolute value decays as O(|t|^{-n/2}) at infinity; we always take the resulting continuous version of its density.

Suppose, to the contrary, that arrays d^{(r)} satisfy


f_{d^{(r)}}(0)to0.


Sort each array in nonincreasing order and extend it by zeros. A diagonal subsequence has


d_i^{(r)}to c_i


for every i, with


sum_i c_i^2le1.


We consider the two possible values of c_1.

##### Case 1: c_1=0

For dge0, define


psi_d(t)=mathbb EbbE e^{itd(G^2-1)}
=e^{-itd}(1-2itd)^{-1/2}.


For logarithms of characteristic functions, use the continuous branch equal to zero at t=0. For fixed t and |td| small, expansion gives


logpsi_d(t)=-t^2d^2+O(|t|^3d^3).


It follows that the characteristic function


phi_r(t)=prod_ipsi_{d_i^{(r)}}(t)


converges pointwise to e^{-t^2}, because


sum_i(d_i^{(r)})^3
le d_1^{(r)}sum_i(d_i^{(r)})^2
=d_1^{(r)}.


Weak convergence alone does not control a density at one point. To pass through Fourier inversion we need an integrable bound uniform in r. Concavity of log(1+x) and


sum_i(d_i^{(r)})^2=1


give


|phi_r(t)|
=prod_i(1+4(d_i^{(r)})^2t^2)^{-1/4}
le
(1+4(d_1^{(r)})^2t^2)^{-1/(4(d_1^{(r)})^2)}.


The function xmapstolog(1+Ax)/x is decreasing for Age0. Once (d_1^{(r)})^2le1/8, the last expression is at most


(1+t^2/2)^{-2}.


Fourier inversion and dominated convergence imply


f_{d^{(r)}}(0)
=frac1{2pi}int_{mathbb R}bbRphi_r(t),dt
→
frac1{2pi}int_{mathbb R}bbRe^{-t^2},dt
=frac1{2sqrtpi},


a contradiction.

##### Case 2: c_1>0

Remove the largest coordinate and write


R_r=sum_{ige2}d_i^{(r)}(G_i^2-1).


We claim that R_r converges in distribution to


R=sum_{ige2}c_i(G_i^2-1)+Z,

Zsim N(0,,2(1-sum_{ige1}c_i^2)),


where Z and the displayed Gaussians are independent.

The series converges in L^2, since the tail beyond m has variance


2sum_{i>m}c_i^2to0.


To identify the limit, note that the sorted weights satisfy


d_{m+1}^{(r)}le(m+1)^{-1/2}.


For fixed t and sufficiently large m, the same logarithmic expansion as above gives, uniformly in r,


logmathbb EbbE e^{itR_r}
=
sum_{i=2}^mlogpsi_{d_i^{(r)}}(t)
-t^2(1-sum_{i=1}^m(d_i^{(r)})^2)
+O(frac{|t|^3}{√(m+1)}).


First let rto∞ and then mto∞. The resulting characteristic function is exactly that of R, which proves the claim.

The residual limit has positive mass arbitrarily close to zero. This is the fact that allows the surviving largest coordinate to control the convolution density.

For any delta>0, choose a finite truncation of its series whose remaining tail variance is less than delta^2/16. Each nonzero summand in the finite part has a density positive near zero, so the finite part together with Z has positive probability in (-delta/2,delta/2). Zero summands or a degenerate Z cause no difficulty. The independent tail has positive probability in the same interval by Chebyshev's inequality. Intersecting these events proves the assertion. This argument uses square summability of (c_i), not summability.

In particular, by convergence in distribution and the open-set inequality,


liminf_{rto∞}mathbb PbbP{|R_r|<c_1/3}
gemathbb PbbP{|R|<c_1/3}>0.


The density of d(G^2-1), for d>0, is


g_d(x)=
frac{e^{-(x+d)/(2d)}}{√(2pi d(x+d))}
mathbf 1_bf1_{{x>-d}}.


For d in a sufficiently small neighborhood of c_1, this density is bounded below by a positive constant depending on c_1 whenever |x|le c_1/3. This interval stays strictly inside its support.

Independence now yields the pointwise convolution identity


f_{d^{(r)}}(0)=mathbb E[g_bbE[g_{d_1^{(r)}}(-R_r)].


To justify it at this particular point, rather than only almost everywhere, the residual R_r has a bounded density by the two-square smoothing bound, using two of its positive coordinates. Convolution of that bounded density with the L^1 function g_{d_1^{(r)}} is continuous: its change under translation by h is bounded by the residual density supremum times


|g_{d_1^{(r)}}(cdot+h)-g_{d_1^{(r)}}|_{L^1},


which tends to zero. It therefore equals the continuous density f_{d^{(r)}} everywhere. Consequently,


liminf_{rto∞}f_{d^{(r)}}(0)
ge c(c_1)liminf_{rto∞}mathbb PbbP{|R_r|<c_1/3}>0,


again a contradiction. The constant may depend on the subsequential limit: any positive liminf contradicts the chosen sequence tending to zero.

The two cases prove a universal lower bound and hence the lemma. QED


---

### 7.4 Partition bounds on the sphere and on hyperplanes

The density-at-the-mean lemma lets us compare S(a) with e^{nB(a)}.

#### Lemma 7.3 (Partition bounds)

There are universal constants c,C>0 such that, for every nge3, every ordered spectrum


lambda_1ge...gelambda_n,

max_i|lambda_i|le3,


and every ain[0,2],


cn^{-1/2}e^{nB(a)}
le S(a)le
Cn^{1/2}e^{nB(a)}.


If Hsubsetmathbb R^nbbR^n is any subspace of dimension n-1, and sigma_{n,H} is uniform probability on


{uin H:|u|^2=n},


then


S_H(a):=int e^{a u^TΛ u/2},sigma_{n,H}(du)
le Cn^{1/2}e^{nB(a)}.


The same constant C works for all H and all ain[0,2].

#### Proof

The chi-square density is


p_{chi_m^2}(r)
=frac{r^{m/2-1}e^{-r/2}}{2^{m/2}Gamma(m/2)},
 r>0.


Stirling's formula gives


cn^{-1/2}le p_{chi_m^2}(n)le Cn^{-1/2}


for m=n or m=n-1 and nge3; the constants absorb the finitely many small dimensions.

By Lemma 7.2 and


(sum_i k_i(a)^2)^{1/2}lesum_i k_i(a)=n,


we have


p_a(n)ge c/n.


By the two-square smoothing bound and k_i(a)ge1/13, we also have


sup_r p_a(r)le C.


Lemma 7.1 proves the two full-space bounds.

We now pass to a hyperplane. Let Λ_H be the compression of Λ to H, and write its eigenvalues as


theta_1ge...getheta_{n-1}.


The min--max principle gives


lambda_igetheta_igelambda_{i+1}.


At the full-space minimizer z=z_a, the matrix


Q_H=z_aI_H-aΛ_H


is positive definite. Since age0, interlacing gives


(det Q_H)^{-1/2}
leprod_{i=1}^{n-1}(z_a-alambda_i)^{-1/2}
=(prod_{i=1}^n(z_a-alambda_i)^{-1/2})
√(z_a-alambda_n).


The last square root is at most √(13).

Let p_{H,a} be the squared-norm density of a Gaussian on H with covariance Q_H^{-1}. The same polar-coordinate calculation as in Lemma 7.1, now in dimension n-1 but still at squared radius n, yields


S_H(a)
=e^{n(z_a-1)/2}(det Q_H)^{-1/2}
frac{p_{H,a}(n)}{p_{chi_{n-1}^2}(n)}.


Every eigenvalue of Q_H is at most


z_a-alambda_nle13,


so all its Gaussian variances are at least 1/13. There are at least two of them, and the two-square smoothing bound gives


p_{H,a}(n)le C.


Combining this bound with the determinant estimate proves the assertion.

Only an upper density bound is used: the compressed Gaussian need not have mean squared radius n, and z_a need not minimize its own variational problem. QED

---

### 7.5 Critical parameters

Now impose Proposition 6.1. Fix a deterministic L_nto∞ with L_nlelog n, put


s_0=L_n n^{-2/3},


and take spectra in the good event mathcal G_n(L_n).calG_n(L_n).

All statements below hold for all sufficiently large n, uniformly over these spectra.

At a=1, abbreviate


k_i=k_i(1),

Y_isim N(0,k_i) independently,

p=p_1.


For each i, let p_{-i} be the density of


sum_{jne i}Y_j^2.


#### Lemma 7.4 (Gaussian parameters at criticality)

With the preceding notation,


lambda_1<z_1<2+s_0,



(sum_i k_i^2)^{1/2}le Cn√(s_0),


and


k_jge c/s_0
(1le jle3).


Moreover,


p(n)gefrac{c}{n√(s_0)},

sup_{rinmathbb R}bbRp_{-i}(r)le Cs_0
(1le ile n).


#### Proof

We compare the exact saddle with the explicit point 2+s_0.

The first trace locates the saddle and controls the total increase in variances when moving to it; the second trace bounds the variances at the explicit point.

Set


h_i=(2+s_0-lambda_i)^{-1},


whose denominators are positive by the edge estimates. Since


R(s)=1-sqrt s+O(s)
(s↓0),


Proposition 6.1 gives, uniformly over the good spectra,


sum_i h_i
=n(1-√(s_0)+o(√(s_0)))<n.


The function


zmapstosum_i(z-lambda_i)^{-1}


is strictly decreasing from infinity on (lambda_1,∞). Its value at z_1 is n; hence


lambda_1<z_1<2+s_0


and


k_ige h_i.


Because sum_i k_i=n, the total increase is exactly


sum_i(k_i-h_i)
=n-sum_i h_i
=n√(s_0)(1+o(1)).


A nonnegative vector has Euclidean norm at most the sum of its coordinates. The triangle inequality and the second resolvent bound therefore give


(sum_i k_i^2)^{1/2}
le(sum_i h_i^2)^{1/2}
+sum_i(k_i-h_i)
le Csqrt n,s_0^{-1/4}+Cn√(s_0)
le Cn√(s_0).


For the last step,


n^{-1/2}s_0^{-3/4}=L_n^{-3/4}to0.


For jle3, the edge estimates also give


0<z_1-lambda_jle2+s_0-lambda_jle Cs_0,


which proves the lower bounds on the first three variances.

Lemma 7.2 now gives the asserted lower bound on p(n).

The three leading variances are needed for a uniform deletion estimate: after deleting any one index i, at least two of the indices 1,2,3 remain. Applying the two-square smoothing bound to their squares, and then convolving with the other squares, yields


sup_r p_{-i}(r)le Cs_0.


QED

---

### 7.6 Small coordinate projections on the tilted sphere

#### Proposition 7.5 (Small coordinate)

For every b>0, the preceding good spectra satisfy


sigma_Λ{|u_1|le bn^{1/3}}
le CL_n^2b.


In particular, given any positive deterministic b_nto0, one can choose L_nto∞ with L_nlelog n and L_n^2b_nto0, so that the probability above tends to zero uniformly over the resulting good spectra.

#### Proof

Conditioning a single coordinate makes the role of the radial estimates transparent. Its unconditioned Gaussian mass is small, and the ratio of the residual radial density to the full radial density quantifies the cost of conditioning.

Let varphi_k be the density of N(0,k). Lemma 7.1 gives, for T>0,


sigma_Λ{|u_1|le T}
=frac1{p(n)}
int_{|y|le T}varphi_{k_1}(y)p_{-1}(n-y^2),dy
le(Cs_0)/(p(n))mathbb PbbP{|Y_1|le T}.


The densities of the nonnegative residual sums are taken to be zero at negative arguments.

By Lemma 7.4 and the Gaussian density bound,


(Cs_0)/(p(n))
le Cns_0^{3/2},


and


mathbb PbbP{|Y_1|le bn^{1/3}}
lefrac{Cbn^{1/3}}{√(k_1)}
le Cbn^{1/3}√(s_0)
=Cb√(L_n).


Their product is CbL_n^2, proving the proposition.

For the final assertion, for example,


L_n=min{log n,b_n^{-1/4}}


for all sufficiently large n has the required properties. QED

---

### 7.7 Coordinate moments and squared overlap

#### Lemma 7.6 (Squared overlap on the sphere)

The probability sigma_Λ has mean zero and diagonal covariance in the eigenbasis of Λ. For every coordinate,


int u_i^2,sigma_Λ(du)
le(Cs_0k_i)/(p(n)).


Consequently, for independent U,V with law sigma_Λ,


mathbb E((U^TV)/(n))^2bbE((U^TV)/(n))^2
le(Cs_0^2)/(n^2p(n)^2)sum_i k_i^2
le Cn^{-2/3}L_n^4.


#### Proof

We reuse the one-coordinate conditioning, now with the weight y^2.

First, the defining density of sigma_Λ is unchanged by changing the sign of any individual coordinate. This proves the assertions about its mean and off-diagonal covariances.

Conditioning as in the preceding proof, now with the nonnegative integrand y^2, gives


int u_i^2,sigma_Λ(du)
=frac1{p(n)}
int_{mathbb R}bbRy^2varphi_{k_i}(y)p_{-i}(n-y^2),dy
le(Cs_0)/(p(n))mathbb EbbE Y_i^2
=(Cs_0k_i)/(p(n)).


Write


m_i=int u_i^2,sigma_Λ(du).


Independence of U,V and diagonality give


mathbb E((U^TV)/(n))^2bbE((U^TV)/(n))^2
=frac1{n^2}sum_i m_i^2
le(Cs_0^2)/(n^2p(n)^2)sum_i k_i^2.


Lemma 7.2 bounds p(n)^{-2} by


Csum_i k_i^2,


and Lemma 7.4 bounds the latter sum by Cn^2s_0. The resulting upper bound is


Cn^2s_0^4=Cn^{-2/3}L_n^4,


as claimed. QED


---

## 8. Transfer from the sphere to the cube

We now transfer spherical estimates to the Ising cube. The bridge is Haar averaging over eigenvector orientations.

Throughout this section, let L_nto∞ with L_nlelog n, put


s_0=L_n n^{-2/3},


and let mathcal G_n=calG_n=mathcal G_n(L_n)calG_n(L_n) be the eigenvalue-only good event of Proposition 6.1.

We fix a deterministic sequence of spectra in mathcal G_ncalG_n and use the notation


Λ=diag(lambda_1,...,lambda_n),

S(a), B(a), z_a, sigma_Λ


from Section 7.

The error bounds supplied by Proposition 6.1 are deterministic on mathcal G_ncalG_n and hold there for every fixed finite upper resolvent parameter.

### 8.1 Haar averaging and the two overlap laws

Let O be Haar distributed on the orthogonal group, independent of the ordered spectrum, and put


W=OΛ O^T.


Orthogonal invariance of GOE justifies this representation: Haar averaging a test function of a symmetric matrix depends only on its eigenvalues, and gives the invariant probability measure on their conjugacy orbit.

We write mathbb E_ObbE_O and mathbb P_ObbP_O for expectation and probability over O with Λ fixed.

Define the normalized Ising partition sum


I=2^{-n}sum_{xin{-1,+1}^n}
exp((x^TOΛ O^Tx)/(2)).


For every fixed cube point x, the vector O^Tx is uniform on the sphere of radius sqrt n. Averaging each summand therefore gives the exact first-moment identity


mathbb E_ObbE_O I=S(1).


To control the normalization of the Gibbs law, we will compare the second moment of I with S(1)^2. Both quantities can be expressed using the same pair integral; only their overlap laws differ.

Let


mathcal Q_n=calQ_n={-1+2j/n:0le jle n},


and


p_I(q)=2^{-n}binom{n}{n(1+q)/2}
(qinmathcal Q_n).calQ_n).


This is the overlap law of two independent uniform cube points.

Let (U,V) be a uniformly oriented orthogonal pair, each vector having norm sqrt n, and define, for -1le qle1,


M(q)=mathbb EbbEexp(
((1+q)U^TΛ U+(1-q)V^TΛ V)/(2)
).


If cube points x,y have overlap qin(-1,1), their normalized sum and difference


frac{x+y}{√(2(1+q))},

frac{x-y}{√(2(1-q))}


are orthogonal and both have norm sqrt n. Rotating them by O^T gives the pair in the definition of M.

At q=1 or q=-1, the points satisfy y=x or y=-x, respectively, and the combined energy is x^TWx; accordingly


M(1)=M(-1)=S(2).


Thus the second-moment identity holds at every lattice point:


mathbb E_ObbE_O I^2
=sum_{qinmathcal Q_n}calQ_np_I(q)M(q).


The exchangeability of U,V shows that M is even and nondecreasing in |q|. More explicitly, if


A=U^TΛ U,

D=V^TΛ V,


then


M(q)=mathbb E[bbE[
e^{(A+D)/2}cosh(frac{q(A-D)}2)
],


whose integrand is nondecreasing in |q|.

For two independent uniform sphere points the overlap has density


f_S(q)=c_n(1-q^2)^{(n-3)/2},

c_n=(Gamma(n/2))/(sqrtpi,Gamma((n-1)/2)),
 -1<q<1.


To see this, fix the first point by rotation and integrate the remaining n-1 coordinates of the second point; the resulting one-coordinate density is proportional to (1-q^2)^{(n-3)/2}, with the displayed normalization given by the beta integral.

Conditional on this overlap, the normalized sum and difference are again the orthogonal pair above. Independence of the two spherical points consequently gives the exact identity


int_{-1}^1 M(q)f_S(q),dq=S(1)^2.


The moment identities reduce concentration to a comparison of the two overlap laws. Because mathbb E_ObbE_O I=S(1), it is enough to prove


sum_{qinmathcal Q_n}calQ_np_I(q)M(q)
le(1+o(1))int_{-1}^1 M(q)f_S(q),dq


with a deterministic o(1) uniform over mathcal G_n.calG_n.

We will use a variational estimate for large overlaps and a direct comparison of the unweighted overlap laws near zero.

First consider the size of the pair integral. By evenness it suffices to take qge0. Conditional on U, the vector V is uniform on the radius-sqrt n sphere in U^perp. The codimension-one bound of Lemma 7.3 controls its integral at parameter 1-q, uniformly in U. The full-space bound then controls the remaining integral at parameter 1+q.

Dividing by the lower bound for S(1)^2 from the same lemma gives


(M(q))/(S(1)^2)
le Cn^2exp(n[B(1+|q|)+B(1-|q|)-2B(1)]).


This remains valid at |q|=1, since the integral with parameter zero equals one.

The cube overlap mass satisfies


p_I(q)le e^{-nq^2/2}.


Indeed, for a mean of n independent uniform signs, the exponential moment bound cosh tle e^{t^2/2} follows by integrating (logcosh t)''le1 twice from zero. Optimizing the resulting Chernoff bound and using symmetry gives the stated bound on each lattice mass.

Thus a margin below q^2/2 in the exponent of the pair bound will suppress large overlaps.

We split the overlap range at


q_0=n^{-7/24}.


The proof uses three scale relations:


nq_0^3=n^{1/8}→∞,

nq_0^4=n^{-1/6}→0,

(s_0)/(q_0^2)=L_n n^{-1/12}→0.


The first turns a cubic variational margin into a negligible tail. The second permits a relative local comparison of the unweighted cube and sphere overlap laws. The third absorbs the spectral error in the cubic estimate; it uses L_nlelog n.

---

### 8.2 The cubic deficit

The next lemma supplies the margin needed in the pair bound for |q|ge q_0.

#### Lemma 8.1 (Cubic deficit)

There is a universal constant c>0 such that, for all sufficiently large n, uniformly for spectra in mathcal G_ncalG_n and q_0le qle1,


B(1+q)+B(1-q)-2B(1)
le(q^2)/(2)-cq^3.


One may take c=1/24.

#### Proof

We estimate the derivative of the spherical variational value on both sides of the critical parameter a=1. Close to that parameter the resolvent estimates give a quantitative margin; away from it a bounded-convergence argument suffices.

The saddle equation in Lemma 7.1 and implicit differentiation show that z_a is continuously differentiable in a. Differentiating the objective at its minimizer gives


B'(a)=frac1{2n}sum_{i=1}^n
(lambda_i)/(z_a-alambda_i),

B'(a)=(z_a-1)/(2a)
(a>0).


The first identity also holds at a=0, with a one-sided derivative at the endpoint. Since the positive weights (z_a-alambda_i)^{-1} sum to n,


|B'(a)|lefrac12max_i|lambda_i|lefrac32
(0le ale2).


For a>0, put r_a=z_a/a and


m_n(s)=frac1nsum_i(2+s-lambda_i)^{-1}.


The saddle equation is equivalently


frac1nsum_i(r_a-lambda_i)^{-1}=a.


At s=s_0, the spectral estimates give


m_n(s_0)=1-√(s_0)+o(√(s_0))<1.


Thus, for a=1+u with 0le ule1, monotonicity of this resolvent gives r_a<2+s_0 and hence


B'(1+u)
le(1+u)/(2)-(u^2)/(2(1+u))+(s_0)/(2).


The upper derivative estimate must be paired with a lower estimate below a=1. The small-overlap range requires the following quantitative comparison.

If


4√(s_0)le ule1/4,


set


t=(u^2)/(1-u).


Then tle1/12 and 0.9tge14.4s_0. The function R from Proposition 6.1 satisfies


R(t)=1-u,


and


-R'(s)=frac{2+s}{2√(s(4+s))}-frac12
ge(c_1)/(sqrt s)
(0<sle1/12)


for a universal c_1>0.

Integrating the last bound from 0.9t to t therefore gives


R(0.9t)-(1-u)ge c_2sqrt tge c_2u.


The error in


m_n(0.9t)=R(0.9t)+o(sqrt t)


is uniform over the indicated range. It is smaller than this fixed positive margin for all sufficiently large n. Consequently m_n(0.9t)>1-u, so monotonicity implies


r_{1-u}ge2+0.9t.


Substitution in the derivative formula yields


B'(1-u)ge(1-u)/(2)-(u^2)/(20(1-u))
(4√(s_0)le ule1/4).


For 0le ule4√(s_0), use instead


r_{1-u}>lambda_1=2+o(s_0)


to obtain


B'(1-u)
ge(1-u)/(2)-(u^2)/(2(1-u))-o(s_0)
ge(1-u)/(2)-Cs_0.


Write


F(q)=B(1+q)+B(1-q)-2B(1).


On 0le ule1/4,


frac1{2(1+u)}-frac1{20(1-u)}gefrac13.


Combining the preceding bounds, and absorbing u^2le16s_0 on the short interval ule4√(s_0) into the error, gives


F'(u)le u-(u^2)/(3)+Cs_0
(0le ule1/4).


Since F(0)=0,


F(q)le(q^2)/(2)-(q^3)/(9)+Cs_0q
(0le qle1/4).


For qge q_0, the last error divided by q^3 is at most


C(s_0)/(q_0^2)
=CL_n n^{-1/12}→0.


Thus the cubic deficit holds on [q_0,1/4], even with c=1/18.

It remains to cover overlaps bounded away from zero. Here a uniform additive error tending to zero is enough.

Fix ain(0,1). The identity


R(a+a^{-1}-2)=a,


the strict monotonicity of R, and the spectral resolvent estimate imply


r_a→ a+a^{-1},

B'(a)→ a/2.


Indeed, evaluating the empirical resolvent at


a+a^{-1}pmdelta,


with delta>0 small enough that the lower point is above 2, traps r_a between these two points for all large n. The upper resolvent parameter needed here is finite for each fixed a. No estimate uniform as a↓0 is used.

The bound |B'|le3/2 now gives, by dominated convergence,


rho_n:=int_0^1|B'(a)-a/2|,da→0.


Integrating the supercritical derivative bound and bounding the subcritical error by rho_n gives, uniformly for 0le qle1,


F(q)
le(q^2)/(2)
-int_0^q(u^2)/(2(1+u)),du+rho_n+(s_0)/(2)
le(q^2)/(2)-(q^3)/(12)+rho_n+(s_0)/(2).


When qge1/4, the cubic term is bounded away from zero, so the last two errors are absorbed with c=1/24. This also covers q=1.

The conclusion must hold uniformly over the entire deterministic spectral event. The estimates in the shrinking range already have that uniformity. The same is true of rho_nto0: otherwise one could select spectra from mathcal G_ncalG_n along a violating subsequence. Their deterministic spectral bounds would still imply the pointwise convergence and bounded domination just proved, a contradiction.

Thus the two ranges give the claimed uniform statement. QED


---

### 8.3 Concentration of the partition sum

#### Proposition 8.2 (Conditional partition concentration)

For spectra in mathcal G_n,calG_n,


mathbb E_ObbE_O I=S(1),


and


sup_{Λinmathcal G_n}calG_n
mathbb E_O[((I)/(S(1))-1)^2]bbE_O[((I)/(S(1))-1)^2]
→0.


In particular, under the GOE law,


(I)/(S(1))to1


in probability.

#### Proof

The first-moment identity was established above. We now bound the exact second moment by treating the two overlap regions separately.

##### Large overlaps

Combining the pair bound, the Chernoff bound p_I(q)le e^{-nq^2/2}, and Lemma 8.1 gives


p_I(q)(M(q))/(S(1)^2)
le Cn^2e^{-cn|q|^3}
(qinmathcal Q_n,calQ_n, |q|ge q_0).


There are at most n+1 lattice points. Since nq_0^3=n^{1/8},


sum_{substack{qinmathcal Q_ncalQ_n |q|ge q_0}}
p_I(q)(M(q))/(S(1)^2)
le Cn^3e^{-cn^{1/8}}
=:tau_n.


In particular, tau_n is smaller than every fixed negative power of n.

##### Central overlaps

The remaining obstacle is that M(q) may vary too rapidly for a derivative-based Riemann-sum estimate. We compare the unweighted overlap laws locally, then move each lattice mass outward, where M is larger.

Let


Delta=2/n.


Stirling's formula, uniformly for |q|<q_0, gives


p_I(q)=
√((2)/(pi n(1-q^2))),
e^{-nJ(q)}(1+O(n^{-1})),


where


J(q)=frac{(1+q)log(1+q)+(1-q)log(1-q)}2.


Here J(q)=q^2/2+O(q^4). Likewise


c_n=√(n/(2pi))(1+O(n^{-1}))


and


log(1-q^2)=-q^2+O(q^4).


Dividing the two expressions shows that


(p_I(q))/(Delta f_S(q))
=exp(O(n^{-1}+q^2+nq^4))
=1+o(1),
 |q|<q_0,


with a uniform relative error, since nq_0^4to0.

Assign to each retained lattice point the outward bin


mathcal B_q=calB_q=

[q,q+Delta],&q>0,
[q-Delta,q],&q<0,
[-Delta/2,Delta/2],&q=0.



For even n, the lattice near zero is


...,-2Delta,-Delta,0,Delta,2Delta,... .


The centered zero bin leaves gaps between Delta/2 and Delta, and between -Delta and -Delta/2.

For odd n, the nearest points are -Delta/2,Delta/2, whose outward bins leave the gap (-Delta/2,Delta/2).

In both cases all bin interiors are disjoint. Every rinmathcal B_qcalB_q satisfies |r|ge|q|, and all these bins lie in


[-q_0-Delta,q_0+Delta]subset(-1,1)


for large n.

On each such bin,


|log(f_S(r))/(f_S(q))|
le Cn(q_0+Delta)Delta
le C(q_0+Delta),


because


(log f_S)'(r)=-(n-3)r/(1-r^2).


It follows from the local-density comparison that there is a deterministic delta_nto0 such that, for every nonnegative even function h nondecreasing in |q|,


p_I(q)h(q)
le(1+delta_n)int_{mathcal B_q}calB_qh(r)f_S(r),dr
(qinmathcal Q_n,calQ_n, |q|<q_0).


Indeed, the integral is at least


Delta h(q)f_S(q)e^{-C(q_0+Delta)},


whereas the lattice mass is at most


(1+e_n)Delta f_S(q)


for a deterministic e_nto0. One can take


1+delta_n=(1+e_n)e^{C(q_0+Delta)}.


This estimate does not depend on how rapidly h varies. The outermost bin may extend beyond q_0, which causes no difficulty because the comparison integral is over the full interval [-1,1].

##### Combining the two regions

Apply the bin comparison with h=M. Because the bin interiors are disjoint, their sum is bounded by the full spherical integral. Together with the tail estimate and the spherical second-moment identity, it yields


(mathbb E_ObbE_O I^2)/(S(1)^2)
le1+delta_n+tau_n.


The first-moment identity therefore gives


mathbb E_O[((I)/(S(1))-1)^2]bbE_O[((I)/(S(1))-1)^2]
ledelta_n+tau_n
→0.


The bin error is independent of the spectrum, and the tail estimate is uniform on mathcal G_ncalG_n by Lemma 8.1. Thus the relative variance bound is uniform there. The unconditional assertion follows by Chebyshev's inequality and mathbb P(bbP(mathcal G_n^c)calG_n^c)to0. QED

---

### 8.4 Small projections on the Ising cube

#### Theorem 8.3 (Small projections in the Ising law)

Let W be the augmented GOE matrix and let v be a unit top eigenvector. For every positive deterministic sequence b_nto0,


mu{x:|v^Tx|le b_nn^{1/3}}
→0
qquadin probability.


The probability here includes the auxiliary diagonal used to define v; the Gibbs measure itself does not depend on that diagonal.

#### Proof

The restricted partition sum has a small first moment, while the unrestricted sum is close to its spherical mean.

We choose the auxiliary spectral event after the prescribed threshold b_n: take L_nto∞ with L_nlelog n and L_n^2b_nto0. For example, for all sufficiently large n one may take


L_n=min{log n,b_n^{-1/4}}.


In the representation W=OΛ O^T, take v=Oe_1, and define the restricted normalized sum


I_b=2^{-n}sum_x
mathbf 1_bf1_{{|v^Tx|le b_nn^{1/3}}}e^{x^TWx/2}.


For every fixed x, O^Tx is uniform on the sphere and its first coordinate is v^Tx. Consequently Proposition 7.5 gives, conditional on any spectrum in mathcal G_n,calG_n,


mathbb E_O(I_b)/(S(1))bbE_O(I_b)/(S(1))
=sigma_Λ{|U_1|le b_nn^{1/3}}
le CL_n^2b_n.


For every fixed ε>0, Markov's inequality and Proposition 8.2 imply


mathbb P_O((I_b)/(I)>ε)bbP_O((I_b)/(I)>ε)
lemathbb P_O((I)/(S(1))<bbP_O((I)/(S(1))<frac12)
+(2)/(ε)mathbb E_O(I_b)/(S(1))bbE_O(I_b)/(S(1))
→0


uniformly over the good spectra.

The ratio I_b/I is the desired Gibbs mass, so adding mathbb P(bbP(mathcal G_n^c)calG_n^c) proves the claim.

GOE has a simple spectrum almost surely: the set of matrices with repeated eigenvalues is the zero set of the nonzero discriminant polynomial and has Lebesgue measure zero, whereas GOE has a density. Thus any choice of the unit top eigenvector differs from Oe_1 only by sign, which leaves the asserted event unchanged. QED

---

### 8.5 Squared overlap on the cube

#### Proposition 8.4 (Squared overlap)

Let


Q_n=mathbb E_bbE_{mu^{ ⊗ 2}}q(x,y)^2.


Then n^{2/3}Q_n is bounded in probability. Equivalently, for every positive deterministic sequence T_nto∞,


mathbb PbbP{n^{2/3}Q_n>T_n}→0.


This assertion concerns the original off-diagonal SK disorder.

#### Proof

To prove tightness rather than a bound with a fixed logarithmic loss, keep L_n free until the tail threshold has been specified.

The same second-moment comparison works with the nonnegative overlap weight q^2. Expanding two Gibbs weights gives the exact identity


mathbb E_O[((I)/(S(1)))^2Q_n]bbE_O[((I)/(S(1)))^2Q_n]
=frac1{S(1)^2}sum_{qinmathcal Q_n}calQ_np_I(q)q^2M(q).


The function q^2M(q) is nonnegative, even, and nondecreasing in |q|.

The central terms therefore satisfy the same bin comparison. The tails are at most tau_n because q^2le1. Moreover,


frac1{S(1)^2}int_{-1}^1q^2M(q)f_S(q),dq
=mathbb E_bbE_{sigma_Λ^{ ⊗ 2}}
((U^TV)/(n))^2.


Here U,V are independent under the tilted spherical law. Lemma 7.6 bounds this expectation by


Cn^{-2/3}L_n^4.


Thus


mathbb E_O[((I)/(S(1)))^2n^bbE_O[((I)/(S(1)))^2n^{2/3}Q_n]
le CL_n^4+n^{2/3}tau_n
le C'L_n^4


for all large n, uniformly on mathcal G_n.calG_n.

The weighted estimate is now available for every admissible L_n. Given an arbitrary deterministic T_nto∞, choose


L_n=min{log n,T_n^{1/8}}


for all sufficiently large n. Then L_nto∞ and


L_n^4/T_nle T_n^{-1/2}to0.


Writing


widehatwidehat I=I/S(1),I=I/S(1),


conditional Markov's inequality gives


mathbb P_O(n^bbP_O(n^{2/3}Q_n>T_n)
lemathbb P_O(bbP_O(widehatwidehat I<1/2)I<1/2)
+frac4{T_n}mathbb E_O[bbE_O[widehatwidehat I^I^{,2}n^{2/3}Q_n]
le o(1)+(CL_n^4)/(T_n).


The first error is uniform on mathcal G_ncalG_n by Proposition 8.2. Adding mathbb P(bbP(mathcal G_n^c)calG_n^c)to0 proves the tail estimate.

The random variable n^{2/3}Q_n has one fixed law for each n throughout this argument: it is a function of the off-diagonal disorder alone. Changing L_n changes only the auxiliary good event and estimates used to prove the assertion, not this law.

To verify explicitly that the tail estimate implies boundedness in probability, suppose tightness failed. Then there would be an ε>0 and increasing indices n_j with


mathbb P(n_j^bbP(n_j^{2/3}Q_{n_j}>j)>ε.


Define a deterministic staircase by T_n=j for n_jle n<n_{j+1}, with an arbitrary positive initial segment. This sequence tends to infinity and contradicts the tail estimate at n=n_j.

Conversely, tightness directly implies the tail estimate by comparing a divergent threshold with any fixed threshold.

Finally, because the Gibbs law is invariant under adding a diagonal to W, integrating out the auxiliary diagonal preserves the asserted probabilities. QED

---

### 8.6 Covariance scale

#### Corollary 8.5 (Covariance scale)

For every positive deterministic sequence M_nto∞,


mathbb PbbP{
frac{n^{2/3}}{M_n}le|Σ|le M_nn^{2/3}
}→1.


#### Proof

The upper and lower bounds use different outputs of the transfer: squared overlap bounds the covariance norm from above, while the top projection bounds it from below.

Spin-flip symmetry gives mu[x]=0, so


Σ=mu[xx^T]


and


Tr(Σ^2)
=sum_{i,j}mu[x_ix_j]^2
=n^2mathbb E_bbE_{mu^{ ⊗ 2}}q(x,y)^2
=n^2Q_n.


Since Σ is positive semidefinite,


|Σ|^2leTr(Σ^2).


The upper bound follows from Proposition 8.4 with threshold T_n=M_n^2.

For the lower bound use Theorem 8.3 with


b_n=M_n^{-1/4}


and put


B_n=mu{|v^Tx|le b_nn^{1/3}}.


Then B_nto0 in probability and


|Σ|ge v^TΣ v
=mu[(v^Tx)^2]
ge b_n^2n^{2/3}(1-B_n).


On B_nle1/2, for all sufficiently large n this is at least


n^{2/3}/(2√(M_n))ge n^{2/3}/M_n.


Both bounds hold simultaneously with probability tending to one by a union bound. The covariance is unchanged by the auxiliary diagonal, so these probabilities are also those of the original SK disorder. QED


---

## 9. Dynamics: from static estimates to slow mixing

We now use the static estimates to prove slow mixing. The core idea is the sign test from the roadmap.

### 9.1 Proof of Theorem A: typical equilibrium starts are slow

We prove the continuous-time statement first. Fix a deterministic sequence t_nge0 with


t_n=o(n^{2/3}).


Choose a positive deterministic b_nto0 such that


frac{t_n}{b_n^2n^{2/3}}→0.


For example,


b_n=(t_n/n^{2/3})^{1/4}+n^{-1}


works. Put


a_n=b_nn^{1/3}.


#### A sign that persists

Separate the auxiliary randomness by writing


W=W_0+Delta,


where W_0 has zero diagonal and Delta is an independent Gaussian diagonal.

The Gibbs law mu_{W_0}=mu_W and the heat-bath transition laws depend only on W_0. The unit top eigenvector


v=v(W_0+Delta)


is allowed to depend on both matrices.

Set


g(x)=v^Tx.


Conditional on W_0,Delta, start the trajectory from mu_{W_0}.

If |g(X_0)|>a_n, failure of the two endpoint projections to have the same nonzero sign requires


|g(X_{t_n})-g(X_0)|ge a_n.


Hence Lemma 5.2 and Markov's inequality give


mathbb P_bbP_{mu_{W_0}}(
g(X_0),g(X_{t_n}) do not have the same nonzero sign
mid W_0,Delta
)
lemu_{W_0}{|g|le a_n}+(2t_n)/(a_n^2).


By Theorem 8.3, the first term tends to zero in probability over W_0,Delta. It also tends to zero in expectation, since it lies in [0,1]. Therefore the deterministic quantity


ε_n
:=mathbb E_bbE_{W_0,Delta}mu_{W_0}{|g|le a_n}
+(2t_n)/(a_n^2)
→0


bounds the average sign-failure probability.

#### Testing each realized state

We now fix the initial configuration before defining its test event.

For x with g(x)ne0, let


H_{v,x}={y:g(y)g(x)>0},


and


r_n(W_0,Delta,x)
=1-(e^{t_nmathcal L}calLmathbf 1_bf1_{H_{v,x}})(x).


For g(x)=0, set


r_n(W_0,Delta,x)=1.


This definition includes every zero projection in the failure event, and the sign-failure estimate implies


mathbb E_bbE_{W_0,Delta}mu_{W_0}[r_n(W_0,Delta,cdot)]
leε_n.


Spin-flip symmetry gives


mu_{W_0}(H_{v,x})le1/2.


If g(x)ne0, testing total variation against this half-space yields


d_{t_n}^{ct}(W_0,x)
gefrac12-r_n(W_0,Delta,x).


When g(x)=0, the following consequence remains valid because r_n=1:


mathbf 1_bf1_{{d_{t_n}^{ct}(W_0,x)le1/4}}
le4r_n(W_0,Delta,x).


#### Removing the auxiliary diagonal

The test half-space depends on Delta, but the bad-state indicator on the left does not. We can therefore integrate the last inequality to obtain


mathbb E_bbE_{W_0}mu_{W_0}
{x:d_{t_n}^{ct}(W_0,x)le1/4}
=
mathbb E_bbE_{W_0,Delta}mu_{W_0}
{x:d_{t_n}^{ct}(W_0,x)le1/4}
le4ε_n.


In particular, for every fixed delta>0,


mathbb P_bbP_{W_0}(
mu_{W_0}{x:d_{t_n}^{ct}(W_0,x)>1/4}<1-delta
)
le(4ε_n)/(delta)
→0.


This is the asserted convergence in probability over the original off-diagonal disorder.

The argument integrates an intrinsic distance; it does not require choosing one auxiliary diagonal that works for a prescribed collection of initial states.

For deterministic integers k_n=o(n^{5/3}), use


b_n=(k_n/n^{5/3})^{1/4}+n^{-1}


and the discrete increment bound. The same proof replaces 2t_n/a_n^2 by


(2k_n)/(na_n^2)


and proves the displayed conclusion with d_{k_n}^{dt}.

These are distances conditional on the realized initial configuration. Averaging the transition law itself over a Gibbs initial configuration would instead leave it equal to mu_{W_0} by stationarity. QED


---

### 9.2 Proof of Theorem B: covariance and linear tests

We now prove the linear inverse Rayleigh bounds. The covariance estimate controls variances of linear functions. To control their inverse Rayleigh quotients, we must also account for sites with small average conditional variance.

We use the standard spin-deletion cavity reweighting. The calculation below uses only exact deletion identities and conditional Gaussian second moments, which hold at the critical temperature.

#### Proposition 9.1 (Linear tests)

With probability tending to one,


|Σ|lemathcal R_calR_{lin}(mu)
le16|Σ|+n^{13/25}.


In particular, n^{-2/3}mathcal R_calR_{lin}(mu) is bounded in probability.

#### Proof

The denominator in the inverse Rayleigh quotient of a linear function is diagonal in its coefficients. We first rewrite the quotient as a weighted covariance norm, then show that the sites with large weights form a sufficiently small exceptional set.

##### Weighted covariance

For each site define its local field and average conditional variance by


g_i(x)=sum_{jne i}W_{ij}x_j,

alpha_i=mu[sech^2g_i].


Thus 0<alpha_ile1, and Lemma 5.1 gives


mathcal D_calD_mu(a^Tx)=sum_ialpha_i a_i^2,

Var_mu(a^Tx)=a^TΣ a.


It follows by the change of variables


b_i=√(alpha_i)a_i


that


mathcal R_calR_{lin}(mu)=|Gamma|,

Gamma=diag(alpha_i^{-1/2})Σ
diag(alpha_i^{-1/2}).


For every nonzero a, the inequalities alpha_ile1 and the nonnegativity of the variance give


(a^TΣ a)/(sum_ialpha_i a_i^2)
ge(a^TΣ a)/(|a|^2).


Taking the supremum proves the lower bound in the proposition.

##### Deleting one spin

To control a site's weight, let nu_i be the Gibbs law on the other n-1 spins after deleting site i and all its incident couplings, without rescaling the remaining couplings. Write barmu_i for the corresponding marginal of mu and set


Z_i=nu_i[cosh g_i],

R_i=mathbb E_bbE_{nu_i ⊗ nu_i}
[(frac{x_{-i}^Ty_{-i}}{n})^2].


Summing over x_i gives the exact identities


(dbarmu_i)/(dnu_i)=(cosh g_i)/(Z_i),


and


alpha_i=frac{nu_i[(cosh g_i)^{-1}]}{Z_i}
ge Z_i^{-2},


where the last step is Jensen's inequality for the reciprocal function.

Since cosh g_ige1, the product marginal comparison and


(q-x_iy_i/n)^2le2q^2+2n^{-2}


imply


R_i
le Z_i^2mathbb E_bbE_{barmu_i ⊗ barmu_i}
[(frac{x_{-i}^Ty_{-i}}n)^2]
le2Z_i^2(Q_n+n^{-2}).


Let mathcal F_icalF_i be the sigma-field generated by the matrix with row and column i deleted. Conditional on mathcal F_i,calF_i, the law nu_i is fixed and the incident couplings are independent N(0,1/n) variables. For a fixed spin configuration, g_i is consequently a centered Gaussian with variance


sigma_{loc}^2=(n-1)/n.


For every fixed integer mge1, Jensen's inequality yields


mathbb E[Z_i^mbbE[Z_i^mmidmathcal F_i]calF_i]
lemathbb E_bbE_{Gsim N(0,sigma_{loc}^2)}[cosh^mG]
le2e^{m^2/2}.


Taking, for example, m=200 and using a union bound gives


mathbb P(bbP(max_iZ_i>n^{1/100})→0.


Proposition 8.4 says that n^{2/3}Q_n is bounded in probability. Combining it with the cavity overlap bound and the max-normalization estimate gives


mathbb P(E_n)→1,bbP(E_n)→1,

E_n={max_iR_ile r_n},

r_n=n^{-7/12}.


For clarity, the exponent slack here is


1/12-1/50=19/300>0.


Indeed, on


max_iZ_ile n^{1/100}


and


Q_nletfrac14n^{-2/3+19/300},


the first term on the right of the cavity overlap bound is at most r_n/2, and its second term is at most r_n/2 for all sufficiently large n.

##### Counting exceptional sites

The uniform polynomial bound on Z_i is not by itself enough. We also need most Z_i to stay below a fixed constant.

Conditional on mathcal F_i,calF_i, for two configurations of the remaining spins put


c=x_{-i}^Ty_{-i}/n.


Their two local fields have variances sigma_{loc}^2 and covariance c, so


mathbb E[coshbbE[cosh g_i(x)cosh g_i(y)midmathcal F_i]calF_i]
=e^{sigma_{loc}^2}cosh c.


It follows that


mathbb E[Z_ibbE[Z_imidmathcal F_i]calF_i]
=e^{sigma_{loc}^2/2},


and


Var(Z_imidmathcal F_i)calF_i)
=e^{sigma_{loc}^2}
mathbb E_bbE_{nu_i ⊗ nu_i}[cosh c-1]
le CR_i,


because |c|le1 and cosh c-1le Cc^2 on this interval.

The event E_n depends on many incident rows. To use the variance estimate with its stated conditioning, introduce the clipped count


N_n=sum_{i=1}^n
mathbf 1_bf1_{{R_ile r_n}}mathbf 1_bf1_{{Z_i>4}}.


The variable R_i is mathcal F_i-measurable,calF_i-measurable, and


4-e^{sigma_{loc}^2/2}ge4-e^{1/2}>0.


Thus conditional Chebyshev gives


mathbb EbbE N_n
=sum_imathbb E[bbE[
mathbf 1_bf1_{{R_ile r_n}}
mathbb P(Z_i>4bbP(Z_i>4midmathcal F_i)]calF_i)]
le Cnr_n=Cn^{5/12}.


On E_n, the clipped count equals |B|, where


B={i:Z_i>4}.


Consequently,


mathbb P(|B|>n^bbP(|B|>n^{1/2})
lemathbb P(E_n^c)+bbP(E_n^c)+mathbb P(N_n>n^bbP(N_n>n^{1/2})
lemathbb P(E_n^c)+Cn^bbP(E_n^c)+Cn^{-1/12}
→0.


No independence between sites, or conditioning on E_n in the Gaussian calculation, is used.

##### Combining the covariance blocks

The probabilistic work is complete. Work on the event


{max_iZ_ile n^{1/100}, |B|le n^{1/2}},


whose probability tends to one by the max-normalization and exceptional-count estimates.

Let G=B^c. On G,


alpha_i^{-1/2}le Z_ile4.


Thus the principal blocks of the positive semidefinite matrix Gamma satisfy


|Gamma_{GG}|le16|Σ|,


and


|Gamma_{BB}|leTrGamma_{BB}
=sum_{iin B}alpha_i^{-1}
le|B|max_iZ_i^2
le n^{13/25}.


Here Σ_{ii}=1 by spin-flip symmetry.

Positivity also controls the cross block: for uinmathbb R^GbbR^G and winmathbb R^B,bbR^B,


|u^TGamma_{GB}w|^2
le(u^TGamma_{GG}u)(w^TGamma_{BB}w).


This follows by applying nonnegativity to the quadratic form at (u,sw) for all real s.

If


a=|Gamma_{GG}|,

b=|Gamma_{BB}|,


it follows that


(u,w)^TGamma(u,w)
le(sqrt a,|u|+sqrt b,|w|)^2
le(a+b)(|u|^2+|w|^2).


The same bound holds when either block is empty. Therefore


|Gamma|le16|Σ|+n^{13/25},


proving the proposition. Corollary 8.5 and 13/25<2/3 give the final assertion. QED

#### Proof of Theorem B

The factor in the linear comparison is fixed, so it can be absorbed while preserving every divergent threshold.

Let M_nto∞ be positive and deterministic. Apply Corollary 8.5 with the divergent threshold M_n/32, and intersect its event with that of Proposition 9.1. With probability tending to one,


frac{32n^{2/3}}{M_n}le|Σ|
lefrac{M_nn^{2/3}}{32},


and


|Σ|lemathcal R_calR_{lin}(mu)
lefrac{M_nn^{2/3}}2+n^{13/25}.


Since 13/25<2/3, the final term is at most M_nn^{2/3}/2 for all sufficiently large n. These inequalities give the claimed common interval for both quantities. QED


---

### 9.3 The subexponential upper bound and Theorem C

We now prove the upper bounds in Theorem C. The strategy is to compare the critical Gibbs measure with one fixed subcritical high-temperature model.

#### Theorem 9.2 (High-temperature spectral gap)

Fix 0<beta<1. Let J be a symmetric matrix with zero diagonal and independent entries


J_{ij}sim N(0,beta^2/n)
(i<j),


and let


nu_J(x)=Z_J^{-1}exp(frac12x^TJx),
 xin{-1,+1}^n.


There is a finite constant C_beta, depending only on beta, such that


mathbb P_JbbP_J{
Var_{nu_J}(f)le C_betamathcal D_calD_{nu_J}(f)
 for every f:{-1,+1}^ntomathbb RbbR
}→1.


Here mathcal D_calD_{nu_J} is the unscaled sum of conditional variances. Consequently, the discrete-time heat-bath chain choosing one uniform site has spectral gap at least 1/(nC_beta) with probability tending to one.

This theorem is supplied by the separate high-temperature companion paper. The constant C_beta is required only for each fixed beta<1: we select beta after the desired exponential tolerance and before taking the limit in n.

#### Lemma 9.3 (Comparison of conditional-variance forms)

Let rho,nu be probability measures with full support on {-1,+1}^n. Suppose


0<mle(rho(x))/(nu(x))le M<∞
qquadfor every x.


If


Var_nu(f)le C_numathcal D_calD_nu(f)
qquadfor every f,


then


Var_rho(f)
le(M)/(m)C_numathcal D_calD_rho(f)
qquadfor every f.


In particular, for


rho(x)propto e^{H(x)},

nu(x)propto e^{beta H(x)},

beta<1,


one may take


M/m=e^{(1-beta)osc,H}.


#### Proof

The comparison works directly with least-squares characterizations; one need not compare the conditional kernels pointwise.

For any full-support probability measure eta,


Var_eta(f)=inf_{cinmathbb R}bbReta[(f-c)^2],


and


mathcal D_calD_eta(f)
=sum_iinf_{h_i}eta[(f-h_i(x_{-i}))^2],


where every h_i ranges independently over all functions of the other coordinates.

For the second identity, conditional expectation gives the orthogonal decomposition


eta[(f-h_i)^2]
=eta[(f-eta[fmid x_{-i}])^2]
+eta[(eta[fmid x_{-i}]-h_i)^2].


The first identity follows by the same decomposition with constants.

The density bounds and the variational identities imply


Var_rho(f)le MVar_nu(f),


and


mathcal D_calD_rho(f)ge mmathcal D_calD_nu(f),


which prove the comparison.

For the exponential densities, rho(x)/nu(x) is a constant times e^{(1-beta)H(x)}; the common normalization cancels in the ratio of its maximum and minimum. QED

#### Proposition 9.4 (Subexponential upper bound)

For every fixed ε>0,


mathbb PbbP{
t_{mix}le e^{ε n},

k_{mix}le e^{ε n}
}→1.


#### Proof

We keep the order of parameters explicit: first the requested exponential tolerance, then a fixed subcritical temperature, then the dimension.

Fix ε>0 and choose betain(0,1) such that


3(1-beta)<ε.


Let


mu_beta(x)propto e^{beta H_W(x)}.


The zero-diagonal part of beta W satisfies exactly the hypotheses of Theorem 9.2; omitting its diagonal changes the Hamiltonian by a constant. With probability tending to one,


Var_{mu_beta}(f)
le C_betamathcal D_calD_{mu_beta}(f)
qquadfor every f.


Intersect this event with |W|le3, whose probability tends to one by Proposition 6.1.

On their intersection,


osc,H_Wle3n,


and


mu_{min}:=min_xmu(x)
ge2^{-n}e^{-osc,H_W}
ge e^{-(3+log2)n}.


Lemma 9.3 now gives


Var_mu(f)le C_nmathcal D_calD_mu(f),

C_n=C_beta e^{3(1-beta)n}.


We may increase C_beta to be at least one.

The comparison has produced a full Poincare inequality. To obtain worst-start total variation for both clocks, combine its spectral decay with the lower bound on the smallest Gibbs mass. We give the conversion explicitly so that the factor n in attempted updates remains visible.

By Lemma 5.1, the Poincare inequality gives a continuous-time gap at least 1/C_n and a discrete-time gap at least 1/(nC_n).

Since P is an average of orthogonal projections, its spectrum is nonnegative. Spectral decomposition on the mean-zero subspace therefore yields


|e^{tmathcal L}calLh|_{L^2(mu)}
le e^{-t/C_n}|h|_{L^2(mu)},


and


|P^kh|_{L^2(mu)}
le e^{-k/(nC_n)}|h|_{L^2(mu)}.


For a starting configuration x, take


h_x(y)=frac{mathbf 1_bf1_{{y=x}}}{mu(x)}-1,


whose squared norm is


mu(x)^{-1}-1.


Reversibility identifies the transition density relative to mu, minus one, with e^{tmathcal L}calLh_x or P^kh_x. The L^1 expression for total variation and Cauchy--Schwarz give


max_x d_t^{ct}(W,x)
letfrac12mu_{min}^{-1/2}e^{-t/C_n},


and


max_x d_k^{dt}(W,x)
letfrac12mu_{min}^{-1/2}e^{-k/(nC_n)}.


With


B_n=log2+tfrac12(3+log2)n,


these bounds imply


t_{mix}le C_nB_n,

k_{mix}lelceil nC_nB_nrceil.


The two right-hand sides grow at most as a fixed constant depending on beta times n^2e^{3(1-beta)n}, plus the rounding term. Since beta is fixed and 3(1-beta)<ε, both are at most e^{ε n} for all sufficiently large n.

The two events used above have an intersection of probability tending to one without any independence assumption. Finally, the mixing times depend only on the off-diagonal disorder, so averaging out the auxiliary diagonal preserves the conclusion. QED

#### Proof of Theorem C

The independent upper comparison is now proved. It remains to extract worst-start lower bounds from the stronger realized-start statement.

Fix ε>0 and apply Theorem A to


t_n=n^{2/3-ε},

k_n=lfloor n^{5/3-ε}rfloor.


For each clock, with probability tending to one there is a starting configuration whose distance at the displayed time exceeds 1/4.

Worst-start distance is nonincreasing in time by contraction of total variation under a Markov kernel. Hence


t_{mix}ge t_n


and


k_{mix}>k_n


on those events. Since k_{mix} is an integer, the latter implies


k_{mix}ge n^{5/3-ε},


including when k_n=0.

Intersect these two lower-bound events with the event in Proposition 9.4 to obtain all four asserted bounds simultaneously. QED


---

## 10. Appendix: why the equilibrium model alone has no mixing time

The main theorems specify both the update rule and the clock. Two elementary constructions show why these are mathematical assumptions, even if the equilibrium law and reversibility are already fixed.

A third calculation distinguishes high-probability asymptotic bounds from essential bounds over every disorder realization.

### Proposition 10.1 (Arbitrary refresh times)

Fix any realization of the SK measure on Ω_n={-1,+1}^n, nge2. For every positive integer r, there is an irreducible, aperiodic, reversible Markov kernel with stationary law mu whose worst-state threshold-1/4 mixing time equals r.

#### Proof

At each step, refresh the whole configuration from mu with probability p and otherwise leave it fixed. In operator notation,


Pi(x,y)=mu(y),

K_p=(1-p)Id+pPi,

0<ple1.


All entries of K_p are positive, and


mu(x)K_p(x,y)=mu(y)K_p(y,x).


The identity Pi^2=Pi gives, for 0<p<1,


K_p^k=Pi+(1-p)^k(Id-Pi),


and


max_x|K_p^k(x,cdot)-mu|_{TV}
=(1-p)^k(1-mu_{min}).


Here


mu_{min}le2^{-n}le1/4.


Choosing


1-p=[frac{1}{4(1-mu_{min})}]^{1/r}


gives distance exactly 1/4 at r and strictly larger at every earlier integer. The case p=1 also has mixing time one. QED

The same freedom remains if transitions are required to change at most one spin. Given the uniform-site heat-bath kernel P, the kernel


(1-p)Id+pP


has the same stationary law, detailed balance, irreducibility, and transition restriction. Starting from a minimum-mass state, its distance after k steps is at least


(1-p)^k-mu_{min},


by the event that no update was attempted. Its mixing time consequently diverges as p↓0. Allowing p to depend on n changes possible growth classes as well.

We next fix the dynamics completely. The remaining distinction is between typical disorder as n grows and all disorder at a fixed dimension. Unbounded Gaussian couplings already separate those notions for two spins.

### Proposition 10.2 (A two-spin example)

For n=2 and the discrete uniform-site heat-bath kernel, the mixing time has no finite deterministic essential upper bound over the Gaussian coupling, although it is finite for every realization.

#### Proof

A large positive coupling keeps the two aligned states persistent.

This can be quantified using the linear observable


g(x)=x_1+x_2.


Put w=W_{12}. Spin-flip symmetry gives mu[g]=0, and conditional expectation computes the eigenvalue exactly:


Pg=theta g,

theta=(1+tanh w)/(2).


Starting at (1,1), the expectation of g after k steps is 2theta^k. Since |g|_∞=2, the dual total-variation bound yields


|P^k((1,1),cdot)-mu|_{TV}
getfrac12theta^k.


For each fixed k, this exceeds 1/4 when w is sufficiently large and positive. Every such Gaussian tail has positive probability.

The chain for any finite w remains finite, irreducible, and aperiodic, so its own mixing time is finite. QED

Proposition 10.2 concerns essential bounds over all disorder realizations at fixed n. It does not contradict the typical-disorder asymptotic statements in Theorem C.

---

## 11. End of proof

The static spectral estimates of Section 6, the spherical estimates of Section 7, and the Haar transfer of Section 8 prove the small-projection theorem and the squared-overlap bound. Section 9 uses those inputs to prove Theorem A, Theorem B, and Theorem C. This completes the proof.