diff --git a/sk_model_critical_slowing_v2.txt b/sk_model_critical_slowing_v2.txt
index 8be60a4..a99c671 100644
--- a/sk_model_critical_slowing_v2.txt
+++ b/sk_model_critical_slowing_v2.txt
@@ -67,9 +67,9 @@ 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),
+L={i=1}^n(P_i-Id),
 
-P_if(x)=mathbb E_bbE_mu[f(x)mid x_{-i}],
+P_if(x)=E_[f(x)mid x_{-i}],
 
 
 where x_{-i} means all spins except x_i.
@@ -83,7 +83,7 @@ P=frac1nsum_{i=1}^n P_i .
 These clocks are related by
 
 
-mathcal L=n(P-Id).calL=n(P-Id).
+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.
@@ -92,10 +92,10 @@ For a starting configuration x, let
 
 
 d_t^{ct}(W,x)
-=|e^{tmathcal L}calL(x,cdot)-mu_W|_{TV},
+=|e^{tL},·)-mu_W|_{TV},
 
 d_k^{dt}(W,x)
-=|P^k(x,cdot)-mu_W|_{TV},
+=|P^k(x,·)-mu_W|_{TV},
 
 
 where total variation is
@@ -115,7 +115,7 @@ t_{mix}
 
 
 k_{mix}
-=min{kinmathbb NbbNcup{0}:max_x d_k^{dt}(W,x)le1/4}.
+=min{kinN{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).
@@ -137,7 +137,7 @@ we have
 
 mu_W{x:d_{t_n}^{ct}(W,x)>1/4}
 → 1
-qquadin probability over W.
+ in probability over W.
 
 
 For every deterministic integer sequence k_nge0 with
@@ -151,7 +151,7 @@ we have
 
 mu_W{x:d_{k_n}^{dt}(W,x)>1/4}
 → 1
-qquadin probability over W.
+ in 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.
@@ -167,20 +167,20 @@ Let
 and let
 
 
-mathcal D_calD_{mu_W}(f)
+D_{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}}
+R_{lin}(mu_W)
+=sup_{ainR^n∖bbR^n∖{0}}
 frac{Var_{mu_W}(a^Tx)}
-{mathcal D_calD_{mu_W}(a^Tx)} .
+{D_{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
+For every positive deterministic sequence M_nto∞, with probability tending to one both |Σ| and R_{lin}(mu_W) lie in
 
 
 [
@@ -196,13 +196,13 @@ The factor M_n can diverge as slowly as we like. The theorem only controls linea
 For every fixed ε>0,
 
 
-mathbb PbbP{
+P{
 n^{2/3-ε}le t_{mix}le e^{ε n}
 }→ 1,
 
 
 
-mathbb PbbP{
+P{
 n^{5/3-ε}le k_{mix}le e^{ε n}
 }→ 1.
 
@@ -217,11 +217,11 @@ Before proving anything hard, we record the bookkeeping that makes every later s
 
 ### 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
+For every function f:{-1,+1}^ntoR, 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
+D_
+=-< f,Lf>_calLf>_mu
 =sum_{i=1}^nmu[(f-P_if)^2]
 =sum_{i=1}^ninf_{h_i}mu[(f-h_i(x_{-i}))^2],
 
@@ -229,7 +229,7 @@ mathcal D_calD_mu(f)
 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),
+< f,(Id-P)f>_mu=frac1nD_,
 
 
 and the spectrum of P lies in [0,1].
@@ -237,7 +237,7 @@ 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,
+D_{i=1}^n a_i^2alpha_i,
 
 alpha_i=mu[sech^2 g_i],
 
@@ -255,23 +255,23 @@ Conditional expectation onto functions of x_{-i} is the orthogonal projection in
 
 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
+Summing over i gives the formula for D_ Averaging instead gives
 
 
 < f,(Id-P)f>_mu
 =frac1nsum_i|f-P_if|_{L^2(mu)}^2
-=frac1nmathcal D_calD_mu(f).
+=frac1nD_
 
 
 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
+For a linear function, everything except a_ix_i is fixed once x_{-i} is fixed. Under mu(·mid 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
+Thus the contribution of site i to D_ is
 
 
 a_i^2mu[Var(a_ix_imid x_{-i})]
@@ -284,19 +284,19 @@ 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
+Fix W and a function f:{-1,+1}^ntoR. 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),
+E_[(f(X_t)-f(X_0))^2]
+le2t,D_,
  tge0,
 
 
 and
 
 
-mathbb E_bbE_mu[(f(Y_k)-f(Y_0))^2]
-le(2k)/(n)mathcal D_calD_mu(f),
+E_[(f(Y_k)-f(Y_0))^2]
+le(2k)/(n)D_,
  k=0,1,2,... .
 
 
@@ -307,11 +307,11 @@ In particular, for g(x)=v^Tx with |v|=1, the right-hand sides are at most 2t and
 Stationarity gives
 
 
-mathbb E_bbE_mu[(f(X_t)-f(X_0))^2]
-=2< f,(Id-e^{tmathcal L}calL)f>_mu .
+E_[(f(X_t)-f(X_0))^2]
+=2< f,(Id-e^{tL}>_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
+The operator -L 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 .
@@ -320,9 +320,9 @@ The operator -mathcal LcalL is self-adjoint and nonnegative. Diagonalize it in a
 Therefore
 
 
-2< f,(Id-e^{tmathcal L}calL)f>_mu
-le2t< f,-mathcal Lf>_calLf>_mu
-=2tmathcal D_calD_mu(f).
+2< f,(Id-e^{tL}>_mu
+le2t< f,-Lf>_calLf>_mu
+=2tD_
 
 
 For the discrete chain, P is self-adjoint with spectrum in [0,1]. The stationary increment is
@@ -340,13 +340,13 @@ For 0lethetale1,
 Also
 
 
-< f,(Id-P)f>_mu=frac1nmathcal D_calD_mu(f).
+< f,(Id-P)f>_mu=frac1nD_
 
 
 This proves the discrete bound. Finally, Lemma 5.1 gives
 
 
-mathcal D_calD_mu(v^Tx)
+D_
 =sum_i v_i^2mu[sech^2 g_i]
 lesum_i v_i^2=1.
 
@@ -375,7 +375,7 @@ be the overlap of two configurations. For independent x,ysimmu,
 
 
 Tr(Σ^2)
-=n^2mathbb E_bbE_{mu ⊗ mu}[q(x,y)^2].
+=n^2E_{mu ⊗ mu}[q(x,y)^2].
 
 
 #### Proof
@@ -391,7 +391,7 @@ Tr(Σ^2)
 For independent x,ysimmu,
 
 
-mathbb E_bbE_{mu ⊗ mu}[q(x,y)^2]
+E_{mu ⊗ mu}[q(x,y)^2]
 =frac1{n^2}sum_{i,j}mu[x_ix_j]^2 .
 
 
@@ -451,13 +451,13 @@ 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
+There are events G_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})
+P( 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<∞,
+for all sufficiently large n. On G_n(L_n), for every fixed M<∞,
 
 
 lambda_j=2+o(s_0)(1le jle3),
@@ -529,14 +529,14 @@ Repeat the construction inside the trailing block, then inside each successive b
 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))
+E-E/(sqrt n))
 leexp(Ct^2/n),
 
 
 and
 
 
-0lesqrt m-mathbb EbbEchi_mle(C)/(sqrt m).
+0lesqrt m-E/(sqrt m).
 
 
 #### Proof
@@ -544,7 +544,7 @@ and
 The Gaussian-square moment-generating function and Chernoff's inequality give
 
 
-mathbb PbbP{chi_m^2ge ma}le(ae^{1-a})^{m/2}
+P{chi_m^2ge ma}le(ae^{1-a})^{m/2}
 (a>1),
 
 
@@ -557,33 +557,33 @@ a-1-log age(sqrt a-1)^2,
 we obtain
 
 
-mathbb PbbP{|chi_m-sqrt m|ge r}le2e^{-r^2/2}
+P{|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)
+sqrt m-E
+=(Var(chi_m))/(sqrt m+E
 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},
+P{|Z|ge r}le4e^{-r^2/8},
 
-mathbb EbbE Z^{2k}le4cdot8^k k!
+E Z^{2k}le4·8^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)
+E e^{t(X-E X)}
+leE e^{tZ}
+=Ecosh(tZ)
 le1+4sum_{kge1}frac{8^k k!t^{2k}}{(2k)!}
 le e^{32t^2}.
 
@@ -635,13 +635,13 @@ Phi_n(x)
  xge0,
 
 
-the following holds. On an event mathcal F_ncalF_n with
+the following holds. On an event F_n with
 
 
-mathbb P(bbP(mathcal F_n^c)calF_n^c)le Cexp(-cL_n^{1/4}),
+P( F_n^c)calF_n^c)le Cexp(-cL_n^{1/4}),
 
 
-we have, simultaneously for every unit vector uinmathbb R^n,bbR^n,
+we have, simultaneously for every unit vector uinR^n,
 
 
 |u^THu|lePhi_n(u^TEu).
@@ -717,7 +717,7 @@ sigma_j=2^js_*,
 and
 
 
-d_j=max{1,lfloorsigma_j^{-1/2}rfloor}.
+d_j=max{1,floor(sigma_j^{-1/2})}.
 
 
 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.
@@ -737,7 +737,7 @@ S_k=xi_1+...+xi_k
 obey
 
 
-mathbb PbbP{max_{kle m}|S_k|ge a}
+P{max_{kle m}|S_k|ge a}
 le2exp(-(na^2)/(Cd_j)).
 
 
@@ -765,7 +765,7 @@ le C2^rnsigma_j^{3/2}
 
 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
+Consequently, for a sufficiently large fixed constant K, there is an event F_n on which every block of both sequences has all its centered partial sums bounded by
 
 
 K√(d_j/n),(h_n+√(j+r)).
@@ -774,7 +774,7 @@ K√(d_j/n),(h_n+√(j+r)).
 Moreover,
 
 
-mathbb P(bbP(mathcal F_n^c)calF_n^c)
+P( 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}
@@ -785,7 +785,7 @@ Here K is chosen so that the factors in j and r form summable geometric series.
 
 ##### 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.
+We now fix the noise event F_n and choose an arbitrary unit vector u.
 
 Select the first j for which
 
@@ -1116,7 +1116,7 @@ We now transfer the quadratic-form bound of Lemma 6.4 and the deterministic reso
 
 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
+On the event F_n from Lemma 6.4, the form bound controls the Rayleigh quotient of
 
 
 E-H=2I-T
@@ -1126,7 +1126,7 @@ between
 
 
 f_-(e):=e-Phi_n(e)
-qquadandqquad
+ and 
 f_+(e):=e+Phi_n(e).
 
 
@@ -1151,14 +1151,14 @@ Therefore
  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).
+Define G_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 F_n implies it, so Lemma 6.2 transfers the probability bound to W. All remaining conclusions are deterministic on G_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
+m=floor( n^{1/3})
 
 
 and the span of the first three discrete Dirichlet sine modes on {1,...,m}, extended by zero.
@@ -1277,7 +1277,7 @@ 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
+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 G_n(L_n) satisfies the estimates for every fixed finite M, without any condition on eigenvector orientation. QED
 
 
 ---
@@ -1305,7 +1305,7 @@ and put
 Let sigma_n be uniform probability measure on
 
 
-{uinmathbb R^n:bbR^n:|u|^2=n}.
+{uinR^n:bbR^n:|u|^2=n}.
 
 
 For 0le ale2, define
@@ -1409,7 +1409,7 @@ 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
+and let Y^{z,a}sim N(0,Q^{-1}). Its density relative to a standard Gaussian vector at yinR^n is
 
 
 (det Q)^{1/2}
@@ -1537,7 +1537,7 @@ We consider the two possible values of c_1.
 For dge0, define
 
 
-psi_d(t)=mathbb EbbE e^{itd(G^2-1)}
+psi_d(t)=E e^{itd(G^2-1)}
 =e^{-itd}(1-2itd)^{-1/2}.
 
 
@@ -1576,7 +1576,7 @@ 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
+The function x|->log(1+Ax)/x is decreasing for Age0. Once (d_1^{(r)})^2le1/8, the last expression is at most
 
 
 (1+t^2/2)^{-2}.
@@ -1586,9 +1586,9 @@ Fourier inversion and dominated convergence imply
 
 
 f_{d^{(r)}}(0)
-=frac1{2pi}int_{mathbb R}bbRphi_r(t),dt
+=frac1{2pi}int_{R},dt
 →
-frac1{2pi}int_{mathbb R}bbRe^{-t^2},dt
+frac1{2pi}int_{R}{-t^2},dt
 =frac1{2sqrtpi},
 
 
@@ -1627,7 +1627,7 @@ 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}
+logE e^{itR_r}
 =
 sum_{i=2}^mlogpsi_{d_i^{(r)}}(t)
 -t^2(1-sum_{i=1}^m(d_i^{(r)})^2)
@@ -1643,8 +1643,8 @@ For any delta>0, choose a finite truncation of its series whose remaining tail v
 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.
+liminf_{rto∞}P{|R_r|<c_1/3}
+geP{|R|<c_1/3}>0.
 
 
 The density of d(G^2-1), for d>0, is
@@ -1660,20 +1660,20 @@ For d in a sufficiently small neighborhood of c_1, this density is bounded below
 Independence now yields the pointwise convolution identity
 
 
-f_{d^{(r)}}(0)=mathbb E[g_bbE[g_{d_1^{(r)}}(-R_r)].
+f_{d^{(r)}}(0)=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},
+|g_{d_1^{(r)}}(·+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,
+ge c(c_1)liminf_{rto∞}P{|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.
@@ -1705,7 +1705,7 @@ 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
+If HsubsetR^n is any subspace of dimension n-1, and sigma_{n,H} is uniform probability on
 
 
 {uin H:|u|^2=n},
@@ -1821,7 +1821,7 @@ 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).
+and take spectra in the good event G_n(L_n).
 
 All statements below hold for all sufficiently large n, uniformly over these spectra.
 
@@ -1865,7 +1865,7 @@ Moreover,
 
 p(n)gefrac{c}{n√(s_0)},
 
-sup_{rinmathbb R}bbRp_{-i}(r)le Cs_0
+sup_{rinR}{-i}(r)le Cs_0
 (1le ile n).
 
 
@@ -1898,7 +1898,7 @@ sum_i h_i
 The function
 
 
-zmapstosum_i(z-lambda_i)^{-1}
+z|->sum_i(z-lambda_i)^{-1}
 
 
 is strictly decreasing from infinity on (lambda_1,∞). Its value at z_1 is n; hence
@@ -1974,13 +1974,13 @@ In particular, given any positive deterministic b_nto0, one can choose L_nto∞
 
 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,
+Let phi_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}.
+int_{|y|le T}phi_{k_1}(y)p_{-1}(n-y^2),dy
+le(Cs_0)/(p(n))P{|Y_1|le T}.
 
 
 The densities of the nonnegative residual sums are taken to be zero at negative arguments.
@@ -1995,7 +1995,7 @@ le Cns_0^{3/2},
 and
 
 
-mathbb PbbP{|Y_1|le bn^{1/3}}
+P{|Y_1|le bn^{1/3}}
 lefrac{Cbn^{1/3}}{√(k_1)}
 le Cbn^{1/3}√(s_0)
 =Cb√(L_n).
@@ -2027,7 +2027,7 @@ 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
+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.
 
@@ -2043,8 +2043,8 @@ Conditioning as in the preceding proof, now with the nonnegative integrand y^2,
 
 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
+int_{R}{k_i}(y)p_{-i}(n-y^2),dy
+le(Cs_0)/(p(n))E Y_i^2
 =(Cs_0k_i)/(p(n)).
 
 
@@ -2057,7 +2057,7 @@ 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
+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.
 
@@ -2089,9 +2089,9 @@ 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.
+and let G_n= 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
+We fix a deterministic sequence of spectra in G_n and use the notation
 
 
 Λ=diag(lambda_1,...,lambda_n),
@@ -2101,7 +2101,7 @@ 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.
+The error bounds supplied by Proposition 6.1 are deterministic on G_n and hold there for every fixed finite upper resolvent parameter.
 
 ### 8.1 Haar averaging and the two overlap laws
 
@@ -2113,7 +2113,7 @@ 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.
+We write E_O and P_O for expectation and probability over O with Λ fixed.
 
 Define the normalized Ising partition sum
 
@@ -2125,7 +2125,7 @@ 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).
+E_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.
@@ -2133,14 +2133,14 @@ To control the normalization of the Gibbs law, we will compare the second moment
 Let
 
 
-mathcal Q_n=calQ_n={-1+2j/n:0le jle n},
+Q_n={-1+2j/n:0le jle n},
 
 
 and
 
 
 p_I(q)=2^{-n}binom{n}{n(1+q)/2}
-(qinmathcal Q_n).calQ_n).
+(qinQ_n).
 
 
 This is the overlap law of two independent uniform cube points.
@@ -2148,7 +2148,7 @@ 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(
+M(q)=E
 ((1+q)U^TΛ U+(1-q)V^TΛ V)/(2)
 ).
 
@@ -2172,8 +2172,8 @@ 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).
+E_O I^2
+=sum_{qinQ_n}
 
 
 The exchangeability of U,V shows that M is even and nondecreasing in |q|. More explicitly, if
@@ -2187,7 +2187,7 @@ D=V^TΛ V,
 then
 
 
-M(q)=mathbb E[bbE[
+M(q)=E[bbE[
 e^{(A+D)/2}cosh(frac{q(A-D)}2)
 ],
 
@@ -2211,14 +2211,14 @@ Conditional on this overlap, the normalized sum and difference are again the ort
 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
+The moment identities reduce concentration to a comparison of the two overlap laws. Because E_O I=S(1), it is enough to prove
 
 
-sum_{qinmathcal Q_n}calQ_np_I(q)M(q)
+sum_{qinQ_n}
 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.
+with a deterministic o(1) uniform over G_n.
 
 We will use a variational estimate for large overlaps and a direct comparison of the unweighted overlap laws near zero.
 
@@ -2269,7 +2269,7 @@ 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,
+There is a universal constant c>0 such that, for all sufficiently large n, uniformly for spectra in G_n and q_0le qle1,
 
 
 B(1+q)+B(1-q)-2B(1)
@@ -2469,7 +2469,7 @@ 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.
+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 G_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
 
@@ -2480,17 +2480,17 @@ Thus the two ranges give the claimed uniform statement. QED
 
 #### Proposition 8.2 (Conditional partition concentration)
 
-For spectra in mathcal G_n,calG_n,
+For spectra in G_n,
 
 
-mathbb E_ObbE_O I=S(1),
+E_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]
+sup_{ΛinG_n}
+E_O[((I)/(S(1))-1)^2]bbE_O[((I)/(S(1))-1)^2]
 →0.
 
 
@@ -2513,13 +2513,13 @@ Combining the pair bound, the Chernoff bound p_I(q)le e^{-nq^2/2}, and Lemma 8.1
 
 p_I(q)(M(q))/(S(1)^2)
 le Cn^2e^{-cn|q|^3}
-(qinmathcal Q_n,calQ_n, |q|ge q_0).
+(qinQ_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}}
+sum_{substack{qinQ_n |q|ge q_0}}
 p_I(q)(M(q))/(S(1)^2)
 le Cn^3e^{-cn^{1/8}}
 =:tau_n.
@@ -2577,7 +2577,7 @@ with a uniform relative error, since nq_0^4to0.
 Assign to each retained lattice point the outward bin
 
 
-mathcal B_q=calB_q=
+B_q=
 
 [q,q+Delta],&q>0,
 [q-Delta,q],&q<0,
@@ -2595,7 +2595,7 @@ The centered zero bin leaves gaps between Delta/2 and Delta, and between -Delta
 
 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
+In both cases all bin interiors are disjoint. Every rinB_q satisfies |r|ge|q|, and all these bins lie in
 
 
 [-q_0-Delta,q_0+Delta]subset(-1,1)
@@ -2621,8 +2621,8 @@ It follows from the local-density comparison that there is a deterministic delta
 
 
 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).
+le(1+delta_n)int_{B_q},dr
+(qinQ_n, |q|<q_0).
 
 
 Indeed, the integral is at least
@@ -2650,19 +2650,19 @@ This estimate does not depend on how rapidly h varies. The outermost bin may ext
 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)
+(E_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]
+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
+The bin error is independent of the spectrum, and the tail estimate is uniform on G_n by Lemma 8.1. Thus the relative variance bound is uniform there. The unconditional assertion follows by Chebyshev's inequality and P( G_n^c)calG_n^c)to0. QED
 
 ---
 
@@ -2675,7 +2675,7 @@ Let W be the augmented GOE matrix and let v be a unit top eigenvector. For every
 
 mu{x:|v^Tx|le b_nn^{1/3}}
 →0
-qquadin probability.
+ in probability.
 
 
 The probability here includes the auxiliary diagonal used to define v; the Gibbs measure itself does not depend on that diagonal.
@@ -2697,10 +2697,10 @@ 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,
+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 G_n,
 
 
-mathbb E_O(I_b)/(S(1))bbE_O(I_b)/(S(1))
+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.
 
@@ -2708,15 +2708,15 @@ 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))
+P_O((I_b)/(I)>ε)bbP_O((I_b)/(I)>ε)
+leP_O((I)/(S(1))<bbP_O((I)/(S(1))<frac12)
++(2)/(ε)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.
+The ratio I_b/I is the desired Gibbs mass, so adding P( 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
 
@@ -2729,13 +2729,13 @@ GOE has a simple spectrum almost surely: the set of matrices with repeated eigen
 Let
 
 
-Q_n=mathbb E_bbE_{mu^{ ⊗ 2}}q(x,y)^2.
+Q_n=E_{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.
+P{n^{2/3}Q_n>T_n}→0.
 
 
 This assertion concerns the original off-diagonal SK disorder.
@@ -2747,8 +2747,8 @@ To prove tightness rather than a bound with a fixed logarithmic loss, keep L_n f
 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).
+E_O[((I)/(S(1)))^2Q_n]bbE_O[((I)/(S(1)))^2Q_n]
+=frac1{S(1)^2}sum_{qinQ_n}
 
 
 The function q^2M(q) is nonnegative, even, and nondecreasing in |q|.
@@ -2757,7 +2757,7 @@ The central terms therefore satisfy the same bin comparison. The tails are at mo
 
 
 frac1{S(1)^2}int_{-1}^1q^2M(q)f_S(q),dq
-=mathbb E_bbE_{sigma_Λ^{ ⊗ 2}}
+=E_{sigma_Λ^{ ⊗ 2}}
 ((U^TV)/(n))^2.
 
 
@@ -2770,12 +2770,12 @@ Cn^{-2/3}L_n^4.
 Thus
 
 
-mathbb E_O[((I)/(S(1)))^2n^bbE_O[((I)/(S(1)))^2n^{2/3}Q_n]
+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.
+for all large n, uniformly on G_n.
 
 The weighted estimate is now available for every admissible L_n. Given an arbitrary deterministic T_nto∞, choose
 
@@ -2798,20 +2798,20 @@ 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]
+P_O(n^{2/3}Q_n>T_n)
+leP_O( I<1/2)I<1/2)
++frac4{T_n}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 first error is uniform on G_n by Proposition 8.2. Adding P( 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.
+P(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.
@@ -2829,7 +2829,7 @@ Finally, because the Gibbs law is invariant under adding a diagonal to W, integr
 For every positive deterministic sequence M_nto∞,
 
 
-mathbb PbbP{
+P{
 frac{n^{2/3}}{M_n}le|Σ|le M_nn^{2/3}
 }→1.
 
@@ -2849,7 +2849,7 @@ and
 
 Tr(Σ^2)
 =sum_{i,j}mu[x_ix_j]^2
-=n^2mathbb E_bbE_{mu^{ ⊗ 2}}q(x,y)^2
+=n^2E_{mu^{ ⊗ 2}}q(x,y)^2
 =n^2Q_n.
 
 
@@ -2957,7 +2957,7 @@ If |g(X_0)|>a_n, failure of the two endpoint projections to have the same nonzer
 Hence Lemma 5.2 and Markov's inequality give
 
 
-mathbb P_bbP_{mu_{W_0}}(
+P_{mu_{W_0}}(
 g(X_0),g(X_{t_n}) do not have the same nonzero sign
 mid W_0,Delta
 )
@@ -2968,7 +2968,7 @@ By Theorem 8.3, the first term tends to zero in probability over W_0,Delta. It a
 
 
 ε_n
-:=mathbb E_bbE_{W_0,Delta}mu_{W_0}{|g|le a_n}
+:=E_{W_0,Delta}mu_{W_0}{|g|le a_n}
 +(2t_n)/(a_n^2)
 →0
 
@@ -2989,7 +2989,7 @@ and
 
 
 r_n(W_0,Delta,x)
-=1-(e^{t_nmathcal L}calLmathbf 1_bf1_{H_{v,x}})(x).
+=1-(e^{t_nL} 1_bf1_{H_{v,x}})(x).
 
 
 For g(x)=0, set
@@ -3001,7 +3001,7 @@ 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)]
+E_{W_0,Delta}mu_{W_0}[r_n(W_0,Delta,·)]
 leε_n.
 
 
@@ -3030,10 +3030,10 @@ le4r_n(W_0,Delta,x).
 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}
+E_{W_0}mu_{W_0}
 {x:d_{t_n}^{ct}(W_0,x)le1/4}
 =
-mathbb E_bbE_{W_0,Delta}mu_{W_0}
+E_{W_0,Delta}mu_{W_0}
 {x:d_{t_n}^{ct}(W_0,x)le1/4}
 le4ε_n.
 
@@ -3041,7 +3041,7 @@ le4ε_n.
 In particular, for every fixed delta>0,
 
 
-mathbb P_bbP_{W_0}(
+P_{W_0}(
 mu_{W_0}{x:d_{t_n}^{ct}(W_0,x)>1/4}<1-delta
 )
 le(4ε_n)/(delta)
@@ -3082,11 +3082,11 @@ We use the standard spin-deletion cavity reweighting. The calculation below uses
 With probability tending to one,
 
 
-|Σ|lemathcal R_calR_{lin}(mu)
+|Σ|leR_{lin}(mu)
 le16|Σ|+n^{13/25}.
 
 
-In particular, n^{-2/3}mathcal R_calR_{lin}(mu) is bounded in probability.
+In particular, n^{-2/3}R_{lin}(mu) is bounded in probability.
 
 #### Proof
 
@@ -3105,7 +3105,7 @@ 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,
+D_ a_i^2,
 
 Var_mu(a^Tx)=a^TΣ a.
 
@@ -3119,7 +3119,7 @@ b_i=√(alpha_i)a_i
 that
 
 
-mathcal R_calR_{lin}(mu)=|Gamma|,
+R_{lin}(mu)=|Gamma|,
 
 Gamma=diag(alpha_i^{-1/2})Σ
 diag(alpha_i^{-1/2}).
@@ -3141,7 +3141,7 @@ To control a site's weight, let nu_i be the Gibbs law on the other n-1 spins aft
 
 Z_i=nu_i[cosh g_i],
 
-R_i=mathbb E_bbE_{nu_i ⊗ nu_i}
+R_i=E_{nu_i ⊗ nu_i}
 [(frac{x_{-i}^Ty_{-i}}{n})^2].
 
 
@@ -3170,12 +3170,12 @@ imply
 
 
 R_i
-le Z_i^2mathbb E_bbE_{barmu_i ⊗ barmu_i}
+le Z_i^2E_{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
+Let F_i be the sigma-field generated by the matrix with row and column i deleted. Conditional on F_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.
@@ -3184,21 +3184,21 @@ 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]
+E[Z_i^mbbE[Z_i^mmidF_i]calF_i]
+leE_{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.
+P(>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,
+P(E_n)→1,bbP(E_n)→1,
 
 E_n={max_iR_ile r_n},
 
@@ -3229,7 +3229,7 @@ the first term on the right of the cavity overlap bound is at most r_n/2, and it
 
 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
+Conditional on F_i, for two configurations of the remaining spins put
 
 
 c=x_{-i}^Ty_{-i}/n.
@@ -3238,23 +3238,23 @@ 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[coshbbE[cosh g_i(x)cosh g_i(y)midF_i]calF_i]
 =e^{sigma_{loc}^2}cosh c.
 
 
 It follows that
 
 
-mathbb E[Z_ibbE[Z_imidmathcal F_i]calF_i]
+E[Z_ibbE[Z_imidF_i]calF_i]
 =e^{sigma_{loc}^2/2},
 
 
 and
 
 
-Var(Z_imidmathcal F_i)calF_i)
+Var(Z_imidF_i)
 =e^{sigma_{loc}^2}
-mathbb E_bbE_{nu_i ⊗ nu_i}[cosh c-1]
+E_{nu_i ⊗ nu_i}[cosh c-1]
 le CR_i,
 
 
@@ -3267,7 +3267,7 @@ 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
+The variable R_i is F_i-measurable,calF_i-measurable, and
 
 
 4-e^{sigma_{loc}^2/2}ge4-e^{1/2}>0.
@@ -3276,10 +3276,10 @@ The variable R_i is mathcal F_i-measurable,calF_i-measurable, and
 Thus conditional Chebyshev gives
 
 
-mathbb EbbE N_n
-=sum_imathbb E[bbE[
+E N_n
+=sum_iE[bbE[
 mathbf 1_bf1_{{R_ile r_n}}
-mathbb P(Z_i>4bbP(Z_i>4midmathcal F_i)]calF_i)]
+P(Z_i>4bbP(Z_i>4midF_i)]calF_i)]
 le Cnr_n=Cn^{5/12}.
 
 
@@ -3292,9 +3292,9 @@ 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}
+P(|B|>n^bbP(|B|>n^{1/2})
+leP(E_n^c)+bbP(E_n^c)+P(N_n>n^bbP(N_n>n^{1/2})
+leP(E_n^c)+Cn^bbP(E_n^c)+Cn^{-1/12}
 →0.
 
 
@@ -3333,7 +3333,7 @@ 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,
+Positivity also controls the cross block: for uinR^G and winR^B,
 
 
 |u^TGamma_{GB}w|^2
@@ -3380,7 +3380,7 @@ lefrac{M_nn^{2/3}}{32},
 and
 
 
-|Σ|lemathcal R_calR_{lin}(mu)
+|Σ|leR_{lin}(mu)
 lefrac{M_nn^{2/3}}2+n^{13/25}.
 
 
@@ -3412,13 +3412,13 @@ nu_J(x)=Z_J^{-1}exp(frac12x^TJx),
 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
+P_J{
+Var_{nu_J}(f)le C_betaD_{nu_J}(f)
+ for every f:{-1,+1}^ntoR
 }→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.
+Here D_{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.
 
@@ -3428,22 +3428,22 @@ 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.
+ for every x.
 
 
 If
 
 
-Var_nu(f)le C_numathcal D_calD_nu(f)
-qquadfor every f,
+Var_nu(f)le C_nuD_
+ for every f,
 
 
 then
 
 
 Var_rho(f)
-le(M)/(m)C_numathcal D_calD_rho(f)
-qquadfor every f.
+le(M)/(m)C_nuD_
+ for every f.
 
 
 In particular, for
@@ -3469,13 +3469,13 @@ The comparison works directly with least-squares characterizations; one need not
 For any full-support probability measure eta,
 
 
-Var_eta(f)=inf_{cinmathbb R}bbReta[(f-c)^2],
+Var_eta(f)=inf_{cinR}[(f-c)^2],
 
 
 and
 
 
-mathcal D_calD_eta(f)
+D_
 =sum_iinf_{h_i}eta[(f-h_i(x_{-i}))^2],
 
 
@@ -3500,7 +3500,7 @@ Var_rho(f)le MVar_nu(f),
 and
 
 
-mathcal D_calD_rho(f)ge mmathcal D_calD_nu(f),
+D_ mD_,
 
 
 which prove the comparison.
@@ -3512,7 +3512,7 @@ For the exponential densities, rho(x)/nu(x) is a constant times e^{(1-beta)H(x)}
 For every fixed ε>0,
 
 
-mathbb PbbP{
+P{
 t_{mix}le e^{ε n},
 
 k_{mix}le e^{ε n}
@@ -3539,8 +3539,8 @@ The zero-diagonal part of beta W satisfies exactly the hypotheses of Theorem 9.2
 
 
 Var_{mu_beta}(f)
-le C_betamathcal D_calD_{mu_beta}(f)
-qquadfor every f.
+le C_betaD_{mu_beta}(f)
+ for every f.
 
 
 Intersect this event with |W|le3, whose probability tends to one by Proposition 6.1.
@@ -3562,7 +3562,7 @@ ge e^{-(3+log2)n}.
 Lemma 9.3 now gives
 
 
-Var_mu(f)le C_nmathcal D_calD_mu(f),
+Var_mu(f)le C_nD_,
 
 C_n=C_beta e^{3(1-beta)n}.
 
@@ -3576,7 +3576,7 @@ By Lemma 5.1, the Poincare inequality gives a continuous-time gap at least 1/C_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)}
+|e^{tL}|_{L^2(mu)}
 le e^{-t/C_n}|h|_{L^2(mu)},
 
 
@@ -3599,7 +3599,7 @@ 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
+Reversibility identifies the transition density relative to mu, minus one, with e^{tL} or P^kh_x. The L^1 expression for total variation and Cauchy--Schwarz give
 
 
 max_x d_t^{ct}(W,x)
@@ -3640,7 +3640,7 @@ Fix ε>0 and apply Theorem A to
 
 t_n=n^{2/3-ε},
 
-k_n=lfloor n^{5/3-ε}rfloor.
+k_n=floor( n^{5/3-ε}).
 
 
 For each clock, with probability tending to one there is a starting configuration whose distance at the displayed time exceeds 1/4.
@@ -3707,7 +3707,7 @@ K_p^k=Pi+(1-p)^k(Id-Pi),
 and
 
 
-max_x|K_p^k(x,cdot)-mu|_{TV}
+max_x|K_p^k(x,·)-mu|_{TV}
 =(1-p)^k(1-mu_{min}).
 
 
@@ -3766,7 +3766,7 @@ 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}
+|P^k((1,1),·)-mu|_{TV}
 getfrac12theta^k.