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Asymptotics of even–even correlations in the Ising model

Abstract

We consider finite-range ferromagnetic Ising models on \(\mathbb {Z}^d\) in the regime \(\beta <\beta _{\mathrm {\scriptscriptstyle c}}\). We analyze the behavior of the prefactor to the exponential decay of \(\mathrm {Cov}(\sigma _A,\sigma _B)\), for arbitrary finite sets A and B of even cardinality, as the distance between A and B diverges.

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Notes

  1. Let us emphasize that these conditions are actually stronger than needed. In particular, the irreducibility condition could be substantially weakened at the cost of (minor) additional technicalities. However, ferromagnetism and short-range interactions (that is, \(J_x\le e^{-c\Vert x\Vert }\) for some \(c>0\)) are real restrictions.

  2. We emphasize that the constructions in [9, 11, 22] are actually explicit (in particular the measures and the coupling discussed here), but we only formulate the results in the form we need for the present work.

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Acknowledgements

The authors gratefully acknowledge the support of the Swiss National Science Foundation through the NCCR SwissMAP.

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Correspondence to Yvan Velenik.

Appendix A: Random walk estimates

Appendix A: Random walk estimates

In this section, we provide some random walk estimates that are needed in the paper. Since we are already losing multiplicative constants when reducing the analysis to non-intersecting random walks, we do not try to get sharp asymptotics, but prefer to provide instead a short self-contained analysis. It should however be noted that the approach in [18] and [26] can be adapted to our present setting and would yield sharp asymptotics (and even information on higher-order corrections). Set

$$\begin{aligned} \psi _d(n) = {\left\{ \begin{array}{ll} n^{-1} &{} \text {when }d=2,\\ \log (1+n)^{-2} &{} \text {when }d=3. \end{array}\right. } \end{aligned}$$

We use the notations introduced in Sect. 2.3.4.

Proposition A.1

Let \((\check{\mathbf{S}}_n)_{n\ge 0}\) denote the random walk introduced in Sect. 2.3. Then, when \(d\in \{2,3\}\), there exist \(0< c_-\le c_+ < \infty \) such that, for all \(n\ge 1\),

$$\begin{aligned} c_- \psi _d(n) \le \check{\mathbb {P}}_{0}\bigl ( \mathcal {Q}_n(0) \bigm \vert \mathcal {P}_n(0) \bigr ) \le c_+ \psi _d(n). \end{aligned}$$

Moreover, for any \(d\ge 4\), there exists \(c_-(d)>0\) such that, for all \(n\ge 1\),

$$\begin{aligned} \check{\mathbb {P}}_{0}\bigl ( \mathcal {Q}_n(0) \bigm \vert \mathcal {P}_n(0) \bigr ) \ge c_-(d). \end{aligned}$$

The remainder of this appendix is devoted to a proof of Proposition A.1. The arguments used below are heavily inspired by those in [13, 18]. To shorten notations, we set, for \(n\in \mathbb {Z}_{\ge 0}\),

$$\begin{aligned} u_n = p_n(0,0),\qquad f_n = q_n(0,0). \end{aligned}$$

Note that the quantity we want to control can then be expressed as \(f_n/u_n\).

1.1 Upper bound on \(p_n(x,y)\)

It follows from the local limit theorem (see [16]) that there exists \(C_p^+\) such that, for all \(x,y\in \mathbb {Z}^{d-1}\), all \(n\ge 1\) and all \(d\ge 2\),

$$\begin{aligned} p_n(x,y) \le C_p^+ n^{-(d-1)/2}. \end{aligned}$$
(18)

1.2 Lower bound on \(p_n(x,y)\)

It follows from the local theorem (see [16]) that, for any \(c>0\), there exist \(C_p^->0\) such that, for all \(x,y\in \mathbb {Z}^{d-1}\) satisfying \(\Vert x-y\Vert \le c\sqrt{n}\), all \(n\ge 1\) and all \(d\ge 2\),

$$\begin{aligned} p_n(x,y) \ge C_p^- n^{-(d-1)/2}. \end{aligned}$$
(19)

1.3 Upper bound on \(r_n\)

Since \( \sum _{m=0}^n u_m r_{n-m} = 1 \) and the sequence \((r_m)_{m\ge 1}\) is decreasing, we have

$$\begin{aligned} 1 \ge r_n \sum _{m=0}^n u_m. \end{aligned}$$

It then follows from (19) (and \(u_0=1\)) that there exists \(C_r^+\) such that, for all \(n\ge 1\),

$$\begin{aligned} r_n \le C_r^+ {\left\{ \begin{array}{ll} n^{-1/2} &{} \text {when }d=2,\\ (1+\log n)^{-1} &{} \text {when }d=3. \end{array}\right. } \end{aligned}$$
(20)

1.4 Lower bound on \(r_n\)

We start with the case \(d=3\). Let \(C>1\) be a large constant (to be chosen later). Since \( \sum _{m=0}^{Cn} u_m r_{Cn-m} = 1 \) and the sequence \((r_m)_{m\ge 1}\) is decreasing (and bounded by \(1\)), we have, using (18),

$$\begin{aligned} r_n \sum _{m=0}^{Cn-n} u_m \ge 1 - \sum _{m=Cn-n+1}^{Cn} u_m \ge 1 - C_p^+ \sum _{m=Cn-n+1}^{Cn} m^{-1} \ge \frac{1}{2}, \end{aligned}$$

once \(C\) is chosen large enough. Since, again by (18), \( \sum _{m=0}^{Cn-n} u_m \le C_p^+\log (Cn) \), we conclude that there exists \(C_r^->0\) such that, for all \(n\ge 1\),

$$\begin{aligned} r_n \ge C_r^- (1+\log n)^{-1}. \end{aligned}$$
(21)

Let us now turn to the case \(d=2\). Proceeding similarly as above, we write

$$\begin{aligned} r_n \sum _{m=0}^{Cn-n} u_m \ge 1 - \sum _{m=Cn-n+1}^{Cn} u_m r_{Cn-m} \ge 1 - C_p^+ ((C-1)n)^{-1/2} \sum _{m=0}^{n-1} r_m. \end{aligned}$$

Using (20) (and \(r_0=1\)), we see that the last term can again be made smaller than \(1/2\) by choosing \(C\) large enough. Of course, \( \sum _{m=0}^{Cn-n} u_m \le C_p^+ (Cn)^{1/2} \) by (18). It thus follows that one can find \(C_r^->0\) such that, for all \(n\ge 1\),

$$\begin{aligned} r_n\ge C_r^- n^{-1/2}. \end{aligned}$$
(22)

1.5 Upper bound on \(f_n\)

Set \(I=\{\frac{n}{4}, \dots , \frac{n}{3}\}\). Let \( \overset{{}_{\rightarrow }}{\tau }= \inf \{k\ge 0\,:\,S_k^\parallel \ge n/4\} \) and \( \overset{{}_{\leftarrow }}{\tau } = \sup \{k\ge 0\,:\,S_k^\parallel \le 3n/4\} \) (Fig. 6). Now, observe that, since the increments of \(\check{\mathbf{S}}\) have exponential tails, there exists \(c>0\) such that

$$\begin{aligned} f_n \le e^{-cn} + \sum _{m,m'\in I} \sum _{u,u'\ne 0} \check{\mathbb {P}}_{0} \bigl ( S_{\overset{{}_{\rightarrow }}{\tau }}^\perp = u, S_{\overset{{}_{\rightarrow }}{\tau }}^\parallel = m, S_{\overset{{}_{\leftarrow }}{\tau }}^\perp = u', S_{\overset{{}_{\leftarrow }}{\tau }}^\parallel = n-m', S_{\tau _0^\perp }^\parallel = n \bigr ). \end{aligned}$$

Applying twice the strong Markov property (once for the walk itself, once for the time-reversed walk), we can write

$$\begin{aligned}&\check{\mathbb {P}}_{0} \bigl ( S_{\overset{{}_{\rightarrow }}{\tau }}^\perp = u, S_{\overset{{}_{\rightarrow }}{\tau }}^\parallel = m, S_{\overset{{}_{\leftarrow }}{\tau }}^\perp = u', S_{\overset{{}_{\leftarrow }}{\tau }}^\parallel = n-m', S_{\tau _0^\perp }^\parallel = n \bigr ) \\&\quad = \check{\mathbb {P}}_{0} \bigl ( S_{\overset{{}_{\rightarrow }}{\tau }}^\perp = u, S_{\overset{{}_{\rightarrow }}{\tau }}^\parallel = m, \tau _0^\perp> \overset{{}_{\rightarrow }}{\tau }\bigr ) \check{\mathbb {P}}_{0} \bigl ( S_{\overset{{}_{\rightarrow }}{\tau }}^\perp = u', S_{\overset{{}_{\rightarrow }}{\tau }}^\parallel = m', \tau _0^\perp > \overset{{}_{\rightarrow }}{\tau }\bigr )\\&\qquad \times q_{n-m'-m}(u,u'). \end{aligned}$$

Now, on the one hand, (18) implies that

$$\begin{aligned} q_{n-m'-m}(u,u') \le p_{n-m'-m}(u,u') \le C_p^+ (n/3)^{-(d-1)/2}. \end{aligned}$$

On the other hand, by (20),

$$\begin{aligned}&\sum _{m\in I}\sum _{u\ne 0} \check{\mathbb {P}}_{0} \bigl ( S_{\overset{{}_{\rightarrow }}{\tau }}^\perp = u, S_{\overset{{}_{\rightarrow }}{\tau }}^\parallel = m, \tau _0^\perp > \overset{{}_{\rightarrow }}{\tau }\bigr ) \le r_{n/4} \\&\le C_r^+ {\left\{ \begin{array}{ll} (n/4)^{-1/2} &{} \text {if } d=2,\\ \bigl (\log (1+n/4)\bigr )^{-1} &{} \text {if } d=3. \end{array}\right. } \end{aligned}$$

Of course, the same applies to the sum over \(m',u'\). Overall, we conclude that there exists \(C_f^+\) such that, for all \(n\ge 1\),

$$\begin{aligned} f_n \le C_f^+ {\left\{ \begin{array}{ll} n^{-3/2} &{} \text {when }d=2,\\ n^{-1}\bigl (\log (1+n)\bigr )^{-2} &{} \text {when }d=3. \end{array}\right. } \end{aligned}$$
(23)
Fig. 6
figure 6

The type of decomposition used in the derivations of the upper and lower bounds on \(f_n\). Various choices are made for the intervals \(I\) and \(I'\)

1.6 Lower bound on \(f_n\)

We start with the case \(d=3\). Our goal is to prove that there exists \(C_f^- > 0\) such that, for all \(n\ge 1\),

$$\begin{aligned} f_n \ge C_f^- n^{-1} \bigl ( \log (1+n) \bigr )^{-2}. \end{aligned}$$
(24)

First, let us set \(M=[n(\log n)^{-3}]\) and \(I=\{M, \dots , 2M\}\). Similarly as we did for the upper bound, let us introduce \( \overset{{}_{\rightarrow }}{\tau }= \inf \{k\ge 0\,:\,S_k^\parallel \ge M\} \) and \( \overset{{}_{\leftarrow }}{\tau } = \sup \{k\ge 0\,:\,S_k^\parallel \le n-M\} \). We can then write

$$\begin{aligned} f_n&\ge \sum _{m,m'\in I} \sum _{u,u'\ne 0} \check{\mathbb {P}}_{0} \bigl ( S_{\overset{{}_{\rightarrow }}{\tau }}^\perp = u, S_{\overset{{}_{\rightarrow }}{\tau }}^\parallel = m, S_{\overset{{}_{\leftarrow }}{\tau }}^\perp = u', S_{\overset{{}_{\leftarrow }}{\tau }}^\parallel = n-m', S_{\tau _0^\perp }^\parallel = n \bigr ) \\&= \sum _{m,m'\in I} \sum _{u,u'\ne 0} \overset{{}_{\rightarrow }}{q}_m(u) \, q_{n-m'-m}(u,u') \, \overset{{}_{\leftarrow }}{q}_{m'}(u'). \end{aligned}$$

where we have introduced \( \overset{{}_{\rightarrow }}{q}_m(u) = \check{\mathbb {P}}_{0} \bigl ( S_{\overset{{}_{\rightarrow }}{\tau }}^\perp = u, S_{\overset{{}_{\rightarrow }}{\tau }}^\parallel = m, \tau _0^\perp > \overset{{}_{\rightarrow }}{\tau }\bigr ) \) and \( \overset{{}_{\leftarrow }}{q}_{m'}(u') = \check{\mathbb {P}}_{0} \bigl ( S_{\overset{{}_{\rightarrow }}{\tau }}^\perp = u', S_{\overset{{}_{\rightarrow }}{\tau }}^\parallel = m', \tau _0^\perp > \overset{{}_{\rightarrow }}{\tau }\bigr ) \). The next observation is that

$$\begin{aligned} 0\le p_\ell (u,u') - q_\ell (u,u')&= \check{\mathbb {P}}_{u}(\tau _0^\perp \le \ell , \mathcal {P}_\ell (u'))\\&\le \sum _{r=1}^{\ell /2} q_r(u,0) p_{\ell -r}(0,u') + \sum _{r=\ell /2}^{\ell -1} p_r(u,0) q_{\ell -r}(0,u'). \end{aligned}$$

Consequently, writing \(N=N(n,m,m')=n-m-m'\),

$$\begin{aligned} \Bigl | \sum _{m,m'\in I} \sum _{u,u'\ne 0}&\overset{{}_{\rightarrow }}{q}_m(u) \, \bigl (q_{N}(u,u')-p_{N}(u,u')\bigr ) \, \overset{{}_{\leftarrow }}{q}_{m'}(u') \Bigr |\\&\le \sum _{m,m'\in I} \sum _{u,u'\ne 0} \sum _{r=1}^{N/2} \overset{{}_{\rightarrow }}{q}_m(0,u) q_r(u,0) p_{N-r}(0,v) \overset{{}_{\leftarrow }}{q}_{m'}(u',0)\\&\qquad + \sum _{m,m'\in I} \sum _{u,u'\ne 0} \sum _{r=N/2}^{N-1} \overset{{}_{\rightarrow }}{q}_m(0,u) p_r(u,0) q_{N-r}(0,v) \overset{{}_{\leftarrow }}{q}_{m'}(u',0). \end{aligned}$$

By symmetry, it suffices to bound the first sum. First, by (18) and the fact that \(r\le N/2\),

$$\begin{aligned} p_{N-r}(0,v) \le 2C_p^+ N^{-1} \le 2 C_p^+ (n-4M)^{-1} \le C n^{-1}. \end{aligned}$$

Second, by (23),

$$\begin{aligned} \sum _{m\in I} \sum _{u\ne 0} \sum _{r=1}^{N/2} \overset{{}_{\rightarrow }}{q}_m(0,u) q_r(u,0) \le \sum _{r=1}^{n} f_{M+r} \le \frac{C_f^+}{(\log M)^2} \sum _{r=1}^n (M+r)^{-1} \le C \frac{\log \log n}{(\log n)^2}. \end{aligned}$$

Finally, by (20)

$$\begin{aligned} \sum _{m'\in I} \sum _{u'\ne 0} \overset{{}_{\leftarrow }}{q}_{m'}(u',0) \le r_M \le C_r^+ (\log M)^{-1} \le C (\log n)^{-1}. \end{aligned}$$
(25)

We conclude that

$$\begin{aligned} \Bigl | \sum _{m,m'\in I} \sum _{u,u'\ne 0} \overset{{}_{\rightarrow }}{q}_m(u) \, \bigl (q_{N}(u,u')-p_{N}(u,u')\bigr ) \, \overset{{}_{\leftarrow }}{q}_{m'}(u') \Bigr |\le C \frac{\log \log n}{n(\log n)^3}, \end{aligned}$$

which is negligible in view of our target estimate.

Now, by the local CLT [16], there exists \(c\) such that

$$\begin{aligned} \bigl | p_\ell (u,u') - \frac{c}{\ell } \bigr | \le \frac{\epsilon }{\ell } + O(\ell ^{-2}\Vert u'-u\Vert ^2), \end{aligned}$$

uniformly in \(u,u'\) for all \(\ell \) large enough. Using \(\Vert u'-u\Vert ^2 \le 2\Vert u\Vert ^2 + 2\Vert u'\Vert ^2\), (25) and

$$\begin{aligned} \sum _{m\in I} \sum _{u\ne 0} \overset{{}_{\rightarrow }}{q}_{m}(u,0) \Vert u\Vert ^2 \le \check{\mathbb {E}}_0(\Vert \check{\mathbf{S}}^\perp _{\tau _M^\parallel }\Vert ^2) \le C M, \end{aligned}$$

we deduce that

$$\begin{aligned} n^{-2} \sum _{m,m'\in I} \sum _{u,u'\ne 0} \overset{{}_{\rightarrow }}{q}_m(0,u) \Vert u'-u\Vert ^2 \overset{{}_{\leftarrow }}{q}_{m'}(u',0) \le C n^{-1}(\log n)^{-3}, \end{aligned}$$

which is also negligible. (24) thus follows from

$$\begin{aligned} \sum _{m,m'\in I} \sum _{u,v\ne 0} \overset{{}_{\rightarrow }}{q}_m(0,u) N^{-1} \overset{{}_{\leftarrow }}{q}_{m'}(u',0) \ge n^{-1} (1-e^{-c M})^2 (r_M)^2 \ge \tfrac{1}{2} (C_r^-)^2 n^{-1}(\log n)^{-2}, \end{aligned}$$

where we used (21) and the exponential tails of the random walk increments.

The same argument applies when \(d\ge 4\). Indeed, proceeding as before, but with \(M\) being now a large constant independent of \(n\), we get

$$\begin{aligned}&\Bigl | \sum _{m,m'\in I} \sum _{u,u'\ne 0} \overset{{}_{\rightarrow }}{q}_m(u) \, \bigl (q_{N}(u,u')-p_{N}(u,u')\bigr ) \, \overset{{}_{\leftarrow }}{q}_{m'}(u') \Bigr |\\&\quad&\le C n^{-(d-1)/2} \sum _{r=M+1}^\infty f_r. \end{aligned}$$

Fix \(\epsilon >0\). Since \(\sum _{r\ge 1} f_r < 1\), the above abound is smaller than \(\epsilon n^{-(d-1)/2}\), provided \(M>M_0(\epsilon )\). Then, again by the local CLT,

$$\begin{aligned} \bigl | p_\ell (u,u') - c\ell ^{-(d-1)/2} \bigr | \le \frac{\epsilon }{\ell ^{(d-1)/2}} + O(\ell ^{-(d+1)/2}\Vert u'-u\Vert ^2), \end{aligned}$$

uniformly in \(u,u'\) for all \(\ell \) large enough. Proceeding as above, the contribution of the second term is seen to be of order \(n^{-(d+1)/2}\) and thus negligible. Therefore, since

$$\begin{aligned} \sum _{m,m'\in I} \sum _{u,u'\ne 0} \overset{{}_{\rightarrow }}{q}_m(0,u) N^{-(d-1)/2} \overset{{}_{\leftarrow }}{q}_{m'}(u',0) \ge C n^{-(d-1)/2} \end{aligned}$$

for some constant \(C>0\), we conclude that there exists \(C_f^->0\) such that

$$\begin{aligned} f_n \ge C_f^- n^{-(d-1)/2}. \end{aligned}$$
(26)

Let us finally turn to the case \(d=2\), for which we need to proceed differently. Fix \(x,y > 0\) such that \(\check{\mathbb {P}}_0(\check{\mathbf{S}}_1=(x,y))=c>0\). Clearly,

$$\begin{aligned} f_n&\ge \sum _{k=2}^n \check{\mathbb {P}}_0( \check{\mathbf{S}}_k=(n,0), \check{\mathbf{S}}_1=(x,y), \check{\mathbf{S}}_{k-1}=(n-x,y), \check{\mathbf{S}}_j^\perp \ge y\; \forall 2\le j\le k-2)\\&\ge c^2 \sum _{k\ge 1} \check{\mathbb {P}}_0(\check{\mathbf{S}}_k=(n-2x,0), \check{\mathbf{S}}_j^\perp \ge 0\; \forall 1\le j\le k-1). \end{aligned}$$

Consider a trajectory \((\check{\mathbf{S}}_j(\omega ))_{j=0}^k\) be such that \(\check{\mathbf{S}}_0(\omega )=0\) and \(\check{\mathbf{S}}_k(\omega )=(n-2x,0)\). Denote by \(\check{X}_j(\omega )=\check{\mathbf{S}}_{j+1}(\omega )-\check{\mathbf{S}}_{j}(\omega )\) the corresponding increments. Let \(j_0 = \min \{i\ge 0\,:\,\check{\mathbf{S}}_i^\perp (\omega )=\min _{0\le j\le k} \check{\mathbf{S}}_k^\perp (\omega )\}\). Define a new trajectory by setting \(\hat{\mathbf{S}}_0(\omega )=0\) and, for \(1\le j\le k\),

$$\begin{aligned} \hat{\mathbf{S}}_j(\omega ) = \sum _{i=0}^{j-1} \check{X}_{(j_0+i) \text { mod } k}(\omega ). \end{aligned}$$

Observe that \(\hat{\mathbf{S}}_0=0\), \(\hat{\mathbf{S}}_k = (n,0)\) and \(\hat{\mathbf{S}}_i\ge 0\) for all \(1\le i\le k-1\), and that the transformation is measure-preserving. Therefore,

$$\begin{aligned} \check{\mathbb {P}}_0(\check{\mathbf{S}}_k=(n-2x,0), \check{\mathbf{S}}_j^\perp \ge 0\; \forall 1\le j\le k-1) \ge k^{-1} \check{\mathbb {P}}_0(\check{\mathbf{S}}_k=(n-2x,0)) \end{aligned}$$

and, therefore, using (19),

$$\begin{aligned} f_n \ge c^2 n^{-1} u_{n-2x} \ge C_f^- n^{-3/2}, \end{aligned}$$
(27)

for some \(C_f^->0\).

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Ott, S., Velenik, Y. Asymptotics of even–even correlations in the Ising model. Probab. Theory Relat. Fields 175, 309–340 (2019). https://doi.org/10.1007/s00440-018-0890-5

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