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
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.
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The authors gratefully acknowledge the support of the Swiss National Science Foundation through the NCCR SwissMAP.
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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
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 \(n\ge 1\),
Moreover, for any \(d\ge 4\), there exists \(c_-(d)>0\) such that, for all \(n\ge 1\),
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 \(n\in \mathbb {Z}_{\ge 0}\),
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 \(d\ge 2\),
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 \(d\ge 2\),
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
It then follows from (19) (and \(u_0=1\)) that there exists \(C_r^+\) such that, for all \(n\ge 1\),
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),
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 \(n\ge 1\),
Let us now turn to the case \(d=2\). Proceeding similarly as above, we write
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 \(n\ge 1\),
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
Applying twice the strong Markov property (once for the walk itself, once for the time-reversed walk), we can write
Now, on the one hand, (18) implies that
On the other hand, by (20),
Of course, the same applies to the sum over \(m',u'\). Overall, we conclude that there exists \(C_f^+\) such that, for all \(n\ge 1\),
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 \(n\ge 1\),
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
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
Consequently, writing \(N=N(n,m,m')=n-m-m'\),
By symmetry, it suffices to bound the first sum. First, by (18) and the fact that \(r\le N/2\),
Second, by (23),
Finally, by (20)
We conclude that
which is negligible in view of our target estimate.
Now, by the local CLT [16], there exists \(c\) such that
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
we deduce that
which is also negligible. (24) thus follows from
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
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,
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
for some constant \(C>0\), we conclude that there exists \(C_f^->0\) such that
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,
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 \(1\le j\le k\),
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,
and, therefore, using (19),
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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DOI: https://doi.org/10.1007/s00440-018-0890-5
Mathematics Subject Classification
- Primary 60K35
- Secondary 82B20
- 82B41
