diff --git a/sk_model_critical_slowing_v2.txt b/sk_model_critical_slowing_v2.txt
index a99c671..097a262 100644
--- a/sk_model_critical_slowing_v2.txt
+++ b/sk_model_critical_slowing_v2.txt
@@ -83,7 +83,7 @@ P=frac1nsum_{i=1}^n P_i .
 These clocks are related by
 
 
-L=n(P-Id).calL=n(P-Id).
+L=n(P-Id).L=n(P-Id).
 
 
 So one unit of continuous time corresponds roughly to n attempted discrete updates, but the exact relation is the displayed identity.
@@ -175,7 +175,7 @@ be the unscaled Dirichlet form. For linear functions a^Tx, define
 
 
 R_{lin}(mu_W)
-=sup_{ainR^n∖bbR^n∖{0}}
+=sup_{ainR^n∖R^n∖{0}}
 frac{Var_{mu_W}(a^Tx)}
 {D_{mu_W}(a^Tx)} .
 
@@ -263,7 +263,7 @@ Summing over i gives the formula for D_ Averaging instead gives
 =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].
+Since each P_i is an orthogonal projection, 0le< f,P_if>_mule< f,f>_mu. Averaging over i gives the same ineuality 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(·mid x_{-i}), the spin x_i has mean tanh g_i and variance
 
@@ -454,7 +454,7 @@ lambda_1gelambda_2ge...gelambda_n .
 There are events G_n(L_n), measurable with respect to the eigenvalues alone, such that
 
 
-P( G_n(L_n)^c)calG_n(L_n)^c)le Cexp(-cL_n^{1/4})
+P( G_n(L_n)^c)G_n(L_n)^c)le Cexp(-cL_n^{1/4})
 
 
 for all sufficiently large n. On G_n(L_n), for every fixed M<∞,
@@ -541,7 +541,7 @@ and
 
 #### Proof
 
-The Gaussian-square moment-generating function and Chernoff's inequality give
+The Gaussian-square moment-generating function and Chernoff's ineuality give
 
 
 P{chi_m^2ge ma}le(ae^{1-a})^{m/2}
@@ -561,7 +561,7 @@ 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,
+This tail bound gives Var(chi_m)le C. By Jensen's ineuality,
 
 
 sqrt m-E
@@ -578,7 +578,7 @@ E Z^{2k}le4·8^k k!
 (kge1).
 
 
-Jensen's inequality conditional on X, followed by the power series of cosh, gives
+Jensen's ineuality conditional on X, followed by the power series of cosh, gives
 
 
 E e^{t(X-E X)}
@@ -638,7 +638,7 @@ Phi_n(x)
 the following holds. On an event F_n with
 
 
-P( F_n^c)calF_n^c)le Cexp(-cL_n^{1/4}),
+P( F_n^c)F_n^c)le Cexp(-cL_n^{1/4}),
 
 
 we have, simultaneously for every unit vector uinR^n,
@@ -680,7 +680,7 @@ e=sum_{i<n}c_i(u_i-u_{i+1})^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.
+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 ineuality directly. Also c_ige1/sqrt2 when ile n/2.
 
 Therefore
 
@@ -694,7 +694,7 @@ 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
+For the second ineuality, 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
@@ -774,7 +774,7 @@ K√(d_j/n),(h_n+√(j+r)).
 Moreover,
 
 
-P( F_n^c)calF_n^c)
+P( F_n^c)F_n^c)
 le Csum_{j,rge0}2^{r+3j/2}
 e^{-cK^2(h_n+√(j+r))^2}
 le Ce^{-c'h_n^2}
@@ -871,7 +871,7 @@ For 1<sigma_j<8, it follows from d_j=1 with a changed universal constant. The se
 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)}.
+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
@@ -1055,7 +1055,7 @@ then factoring the quadratic gives
 
 
 0ledelta(s)le
-frac{T_D(s)}{n√(s(4+s))}.
+T_D(s)/n√(s(4+s)).
 
 
 Termwise comparison of resolvents shows that
@@ -1069,8 +1069,8 @@ Differentiating the explicit expression for R gives
 
 
 0<-R'(s)
-=frac{2}{√(s(4+s)),
-(2+s+√(s(4+s)))}
+=2/√(s(4+s)),
+(2+s+√(s(4+s)))
 lefrac1{2sqrt s}.
 
 
@@ -1084,7 +1084,7 @@ Hence
 
 
 T_D(s)le(C)/(sqrt s)
-+frac{2T_D(s)}{ns√(s(4+s))}.
++2T_D(s)/ns√(s(4+s)).
 
 
 For sge s_0, the coefficient of the last term is at most
@@ -1093,7 +1093,7 @@ 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.
+Absorbing it proves the bound on T_D. The displayed bound on delta then follows from the root-difference ineuality. 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.
 
@@ -1151,7 +1151,7 @@ Therefore
  1le ile 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).
+Define G_n(L_n) to be precisely the event on which this ineuality 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
 
@@ -1182,7 +1182,7 @@ 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
+The comparison ineualities and Phi_n(0)le Ceta_ns_0 imply
 
 
 max_{jle3}(|lambda_j-2|)/(s_0)
@@ -1305,7 +1305,7 @@ and put
 Let sigma_n be uniform probability measure on
 
 
-{uinR^n:bbR^n:|u|^2=n}.
+{uinR^n:R^n:|u|^2=n}.
 
 
 For 0le ale2, define
@@ -1631,16 +1631,16 @@ logE 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)}).
++O(|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.
+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 ineuality. 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,
+In particular, by convergence in distribution and the open-set ineuality,
 
 
 liminf_{rto∞}P{|R_r|<c_1/3}
@@ -1652,7 +1652,7 @@ 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}}.
+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.
@@ -1863,7 +1863,7 @@ k_jge c/s_0
 Moreover,
 
 
-p(n)gefrac{c}{n√(s_0)},
+p(n)gec/n√(s_0),
 
 sup_{rinR}{-i}(r)le Cs_0
 (1le ile n).
@@ -1921,7 +1921,7 @@ sum_i(k_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
+A nonnegative vector has Euclidean norm at most the sum of its coordinates. The triangle ineuality and the second resolvent bound therefore give
 
 
 (sum_i k_i^2)^{1/2}
@@ -2089,7 +2089,7 @@ Throughout this section, let L_nto∞ with L_nlelog n, put
 s_0=L_n n^{-2/3},
 
 
-and let G_n= 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)G_n(L_n) be the eigenvalue-only good event of Proposition 6.1.
 
 We fix a deterministic sequence of spectra in G_n and use the notation
 
@@ -2156,9 +2156,9 @@ M(q)=E
 If cube points x,y have overlap qin(-1,1), their normalized sum and difference
 
 
-frac{x+y}{√(2(1+q))},
+x+y/√(2(1+q)),
 
-frac{x-y}{√(2(1-q))}
+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.
@@ -2187,7 +2187,7 @@ D=V^TΛ V,
 then
 
 
-M(q)=E[bbE[
+M(q)=E[E[
 e^{(A+D)/2}cosh(frac{q(A-D)}2)
 ],
 
@@ -2347,7 +2347,7 @@ R(t)=1-u,
 and
 
 
--R'(s)=frac{2+s}{2√(s(4+s))}-frac12
+-R'(s)=2+s/2√(s(4+s))-frac12
 ge(c_1)/(sqrt s)
 (0<sle1/12)
 
@@ -2490,7 +2490,7 @@ and
 
 
 sup_{ΛinG_n}
-E_O[((I)/(S(1))-1)^2]bbE_O[((I)/(S(1))-1)^2]
+E_O[((I)/(S(1))-1)^2]E_O[((I)/(S(1))-1)^2]
 →0.
 
 
@@ -2657,12 +2657,12 @@ le1+delta_n+tau_n.
 The first-moment identity therefore gives
 
 
-E_O[((I)/(S(1))-1)^2]bbE_O[((I)/(S(1))-1)^2]
+E_O[((I)/(S(1))-1)^2]E_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 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
+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 ineuality and P( G_n^c)G_n^c)to0. QED
 
 ---
 
@@ -2694,29 +2694,29 @@ In the representation W=OΛ O^T, take v=Oe_1, and define the restricted normaliz
 
 
 I_b=2^{-n}sum_x
-mathbf 1_bf1_{{|v^Tx|le b_nn^{1/3}}}e^{x^TWx/2}.
+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 G_n,
 
 
-E_O(I_b)/(S(1))bbE_O(I_b)/(S(1))
+E_O(I_b)/(S(1))E_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
+For every fixed ε>0, Markov's ineuality and Proposition 8.2 imply
 
 
-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))
+P_O((I_b)/(I)>ε)P_O((I_b)/(I)>ε)
+leP_O((I)/(S(1))<P_O((I)/(S(1))<frac12)
++(2)/(ε)E_O(I_b)/(S(1))E_O(I_b)/(S(1))
 →0
 
 
 uniformly over the good spectra.
 
-The ratio I_b/I is the desired Gibbs mass, so adding P( 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)G_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
 
@@ -2747,7 +2747,7 @@ 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
 
 
-E_O[((I)/(S(1)))^2Q_n]bbE_O[((I)/(S(1)))^2Q_n]
+E_O[((I)/(S(1)))^2Q_n]E_O[((I)/(S(1)))^2Q_n]
 =frac1{S(1)^2}sum_{qinQ_n}
 
 
@@ -2770,7 +2770,7 @@ Cn^{-2/3}L_n^4.
 Thus
 
 
-E_O[((I)/(S(1)))^2n^bbE_O[((I)/(S(1)))^2n^{2/3}Q_n]
+E_O[((I)/(S(1)))^2n^E_O[((I)/(S(1)))^2n^{2/3}Q_n]
 le CL_n^4+n^{2/3}tau_n
 le C'L_n^4
 
@@ -2792,19 +2792,19 @@ L_n^4/T_nle T_n^{-1/2}to0.
 Writing
 
 
-widehatwidehat I=I/S(1),I=I/S(1),
+Ihat=I/S(1),I=I/S(1),
 
 
-conditional Markov's inequality gives
+conditional Markov's ineuality gives
 
 
 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]
++frac4{T_n}E_O[E_O[Ihat^I^{,2}n^{2/3}Q_n]
 le o(1)+(CL_n^4)/(T_n).
 
 
-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 first error is uniform on G_n by Proposition 8.2. Adding P( G_n^c)G_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.
 
@@ -2954,7 +2954,7 @@ If |g(X_0)|>a_n, failure of the two endpoint projections to have the same nonzer
 |g(X_{t_n})-g(X_0)|ge a_n.
 
 
-Hence Lemma 5.2 and Markov's inequality give
+Hence Lemma 5.2 and Markov's ineuality give
 
 
 P_{mu_{W_0}}(
@@ -3021,13 +3021,13 @@ 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}}
+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
+The test half-space depends on Delta, but the bad-state indicator on the left does not. We can therefore integrate the last ineuality to obtain
 
 
 E_{W_0}mu_{W_0}
@@ -3125,7 +3125,7 @@ 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
+For every nonzero a, the ineualities alpha_ile1 and the nonnegativity of the variance give
 
 
 (a^TΣ a)/(sum_ialpha_i a_i^2)
@@ -3158,7 +3158,7 @@ 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.
+where the last step is Jensen's ineuality for the reciprocal function.
 
 Since cosh g_ige1, the product marginal comparison and
 
@@ -3181,10 +3181,10 @@ Let F_i be the sigma-field generated by the matrix with row and column i deleted
 sigma_{loc}^2=(n-1)/n.
 
 
-For every fixed integer mge1, Jensen's inequality yields
+For every fixed integer mge1, Jensen's ineuality yields
 
 
-E[Z_i^mbbE[Z_i^mmidF_i]calF_i]
+E[Z_i^mbbE[Z_i^mmidF_i]F_i]
 leE_{Gsim N(0,sigma_{loc}^2)}[cosh^mG]
 le2e^{m^2/2}.
 
@@ -3198,7 +3198,7 @@ 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
 
 
-P(E_n)→1,bbP(E_n)→1,
+P(E_n)→1,P(E_n)→1,
 
 E_n={max_iR_ile r_n},
 
@@ -3220,7 +3220,7 @@ max_iZ_ile n^{1/100}
 and
 
 
-Q_nletfrac14n^{-2/3+19/300},
+Q_nle1/4n^{-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.
@@ -3238,14 +3238,14 @@ c=x_{-i}^Ty_{-i}/n.
 Their two local fields have variances sigma_{loc}^2 and covariance c, so
 
 
-E[coshbbE[cosh g_i(x)cosh g_i(y)midF_i]calF_i]
+E[coshbbE[cosh g_i(x)cosh g_i(y)midF_i]F_i]
 =e^{sigma_{loc}^2}cosh c.
 
 
 It follows that
 
 
-E[Z_ibbE[Z_imidF_i]calF_i]
+E[Z_ibbE[Z_imidF_i]F_i]
 =e^{sigma_{loc}^2/2},
 
 
@@ -3264,10 +3264,10 @@ The event E_n depends on many incident rows. To use the variance estimate with i
 
 
 N_n=sum_{i=1}^n
-mathbf 1_bf1_{{R_ile r_n}}mathbf 1_bf1_{{Z_i>4}}.
+1_bf1_{{R_ile r_n}}1_bf1_{{Z_i>4}}.
 
 
-The variable R_i is F_i-measurable,calF_i-measurable, and
+The variable R_i is F_i-measurable,F_i-measurable, and
 
 
 4-e^{sigma_{loc}^2/2}ge4-e^{1/2}>0.
@@ -3277,9 +3277,9 @@ Thus conditional Chebyshev gives
 
 
 E N_n
-=sum_iE[bbE[
-mathbf 1_bf1_{{R_ile r_n}}
-P(Z_i>4bbP(Z_i>4midF_i)]calF_i)]
+=sum_iE[E[
+1_bf1_{{R_ile r_n}}
+P(Z_i>4bbP(Z_i>4midF_i)]F_i)]
 le Cnr_n=Cn^{5/12}.
 
 
@@ -3292,9 +3292,9 @@ B={i:Z_i>4}.
 Consequently,
 
 
-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}
+P(|B|>n^P(|B|>n^{1/2})
+leP(E_n^c)+P(E_n^c)+P(N_n>n^P(N_n>n^{1/2})
+leP(E_n^c)+Cn^P(E_n^c)+Cn^{-1/12}
 →0.
 
 
@@ -3384,7 +3384,7 @@ and
 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
+Since 13/25<2/3, the final term is at most M_nn^{2/3}/2 for all sufficiently large n. These ineualities give the claimed common interval for both quantities. QED
 
 
 ---
@@ -3569,9 +3569,9 @@ 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.
+The comparison has produced a full Poincare ineuality. 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).
+By Lemma 5.1, the Poincare ineuality 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
 
@@ -3590,7 +3590,7 @@ 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,
+h_x(y)=frac{1_bf1_{{y=x}}}{mu(x)}-1,
 
 
 whose squared norm is
@@ -3603,20 +3603,20 @@ Reversibility identifies the transition density relative to mu, minus one, with
 
 
 max_x d_t^{ct}(W,x)
-letfrac12mu_{min}^{-1/2}e^{-t/C_n},
+le1/2mu_{min}^{-1/2}e^{-t/C_n},
 
 
 and
 
 
 max_x d_k^{dt}(W,x)
-letfrac12mu_{min}^{-1/2}e^{-k/(nC_n)}.
+le1/2mu_{min}^{-1/2}e^{-k/(nC_n)}.
 
 
 With
 
 
-B_n=log2+tfrac12(3+log2)n,
+B_n=log2+1/2(3+log2)n,
 
 
 these bounds imply
@@ -3767,7 +3767,7 @@ Starting at (1,1), the expectation of g after k steps is 2theta^k. Since |g|_∞
 
 
 |P^k((1,1),·)-mu|_{TV}
-getfrac12theta^k.
+ge1/2theta^k.
 
 
 For each fixed k, this exceeds 1/4 when w is sufficiently large and positive. Every such Gaussian tail has positive probability.