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Register a new userContour methods for long-range Ising models: weakening nearest-neighbor interactions and adding decaying fields
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We consider ferromagnetic long-range Ising models which display phase transitions. They are long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)equiv frac{1}{|x-y|^{2-alpha}}$ with $alpha in [0, 1)$, in particular, $J(1)=1$. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Frohlich-Spencer contours for $alpha eq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Frohlich and Spencer for $alpha=0$ and conjectured by Cassandro et al for the region they could treat, $alpha in (0,alpha_{+})$ for $alpha_+=log(3)/log(2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)gg1$, we can show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any $alpha in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x= h_*cdot(1+|x|)^{-gamma}$ and $gamma >max{1-alpha, 1-alpha^* }$ where $alpha^*approx 0.2714$, the transition still persists.
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Inspired by Fr{o}hlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on $mathbb{Z}^d$, $dgeq 2$. The argument, which is based on a multi-scale analysis, works for the sharp region $alpha>d$ and improves previous results obtained by Park for $alpha>3d+1$, and by Ginibre, Grossmann, and Ruelle for $alpha> d+1$, where $alpha$ is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomial decaying magnetic field with power $delta>0$ as $h^*|x|^{-delta}$, where $h^* >0$. For $d<alpha<d+1$, the phase transition occurs when $delta>d-alpha$, and when $h^*$ is small enough over the critical line $delta=d-alpha$. For $alpha geq d+1$, $delta>1$ it is enough to prove the phase transition, and for $delta=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.
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.
In this paper, we convert the lattice configurations into networks with different modes of links and consider models on networks with arbitrary numbers of interacting particle-pairs. We solve the Heisenberg model by revealing the relation between the Casimir operator of the unitary group and the conjugacy-class operator of the permutation group. We generalize the Heisenberg model by this relation and give a series of exactly solvable models. Moreover, by numerically calculating the eigenvalue of Heisenberg models and random walks on network with different numbers of links, we show that a system on lattice configurations with interactions between more particle-pairs have higher degeneracy of eigenstates. The highest degeneracy of eigenstates of a lattice model is discussed.
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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