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Register a new userCritical two-point function for long-range $O(n)$ models below the upper critical dimension
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We consider the $n$-component $|varphi|^4$ lattice spin model ($n ge 1$) and the weakly self-avoiding walk ($n=0$) on $mathbb{Z}^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $r$ as $r^{-(d+alpha)}$ with $alpha in (0,2)$. The upper critical dimension is $d_c=2alpha$. For $epsilon >0$, and $alpha = frac 12 (d+epsilon)$, the dimension $d=d_c-epsilon$ is below the upper critical dimension. For small $epsilon$, weak coupling, and all integers $n ge 0$, we prove that the two-point function at the critical point decays with distance as $r^{-(d-alpha)}$. This sticking of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.
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Consider the long-range models on $mathbb{Z}^d$ of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $|x|^{-d-alpha}$ for some $alpha>0$. In the previous work (Ann. Probab., 43, 639--681, 2015), we have shown in a unified fashion for all $alpha e2$ that, assuming a bound on the derivative of the $n$-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $G_{p_c}(x)$ decays as $|x|^{alphawedge2-d}$ above the upper-critical dimension $d_cequiv(alphawedge2)m$, where $m=2$ for self-avoiding walk and the Ising model and $m=3$ for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the $n$-step distribution, that $G_{p_c}(x)$ for the marginal case $alpha=2$ decays as $|x|^{2-d}/log|x|$ whenever $dge d_c$ (with a large spread-out parameter $L$). This solves the conjecture in the previous work, extended all the way down to $d=d_c$, and confirms a part of predictions in physics (Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855--868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.
We consider long-range self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)asymp|x|^{-d-alpha}$ with $alpha>0$. The upper-critical dimension $d_{mathrm{c}}$ is $2(alphawedge2)$ for self-avoiding walk and the Ising model, and $3(alphawedge2)$ for percolation. Let $alpha e2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{alphawedge2-d}$, where the constant $Cin(0,infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $alpha<2$ and $alpha>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.
Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.
This is a short review of the two papers on the $x$-space asymptotics of the critical two-point function $G_{p_c}(x)$ for the long-range models of self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$, defined by the translation-invariant power-law step-distribution/coupling $D(x)propto|x|^{-d-alpha}$ for some $alpha>0$. Let $S_1(x)$ be the random-walk Green function generated by $D$. We have shown that $bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($alpha>2$) to Riesz ($alpha<2$), with log correction at $alpha=2$; $bullet~~G_{p_c}(x)simfrac{A}{p_c}S_1(x)$ as $|x|toinfty$ in dimensions higher than (or equal to, if $alpha=2$) the upper critical dimension $d_c$ (with sufficiently large spread-out parameter $L$). The model-dependent $A$ and $d_c$ exhibit crossover at $alpha=2$. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power functions (with log corrections, if $alpha=2$) to optimally control the lace-expansion coefficients $pi_p^{(n)}$, and (iii) probabilistic interpretation (valid only when $alphale2$) of the convolution of $D$ and a function $varPi_p$ of the alternating series $sum_{n=0}^infty(-1)^npi_p^{(n)}$. We outline the proof, emphasizing the above key elements for percolation in particular.
In this paper, we investigate the behaviour of statistical physics models on a book with pages that are isomorphic to half-planes. We show that even for models undergoing a continuous phase transition on $mathbb Z^2$, the phase transition becomes discontinuous as soon as the number of pages is sufficiently large. In particular, we prove that the Ising model on a three pages book has a discontinuous phase transition (if one allows oneself to consider large coupling constants along the line on which pages are glued). Our work confirms predictions in theoretical physics which relied on renormalization group, conformal field theory and numerics ([Car91,ITB91,SMP10]) some of which were motivated by the analysis of the Renyi entropy of certain quantum spin systems.
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