R. Cerf, R. J. Messikh
We study the behavior of the two-dimensional Ising model in a finite box at temperatures that are below, but very close to, the critical temperature. In a regime where the temperature approaches the critical point and, simultaneously, the size of the box grows fast enough, we establish a large deviation principle that proves the appearance of a round Wulff crystal.
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The understanding of site percolation on the triangular lattice progressed greatly in the last decade. Smirnov proved conformal invariance of critical percolation, thus paving the way for the construction of its scaling limit. Recently, the scaling limit of near-critical percolation was also constructed by Garban, Pete and Schramm. The aim of this very modest contribution is to explain how these results imply the convergence, as p tends to p_c, of the Wulff crystal to a Euclidean disk. The main ingredient of the proof is the rotational invariance of the scaling limit of near-critical percolati…
Limit of the Wulff crystal when approaching criticality for isoperimetry in 2D percolation
本文研究三角格点超临界点渗流无穷团簇的等周集，通过调整已有证明方法，证明重标度等周集收敛于归一化Wulff晶体的平移，且趋近临界点时Wulff晶体趋近欧氏圆盘，解决了相关点渗流版本猜想。
The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality
本文研究了FK渗流在q>4接近连续临界点时的行为，证明了相关长度在极限情况下变得各向同性。
We study the surface tension and the phenomenon of phase coexistence for the Ising model on $\mathbbm{Z}^d$ ($d \geqslant 2$) with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations : upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value $τ^q$ of surface tension to maximal flows (first passage times if $d = 2$). For a broad class of distributions of the coupli…
For the two dimensional Ising model, we construct the adequate surface tension near criticality. The latter quantity has been shown to play a central role in the study of phase coexistence in a joint limit where the temperature approaches the critical point from below and simultaneously the size of the system increases fast enough.
We study the 2d-Ising model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FK-percolation events that concern the phenomenon of phase coexistence.
An extension of the Ising-Curie-Weiss model of self-organized criticality with long range interactions
本文扩展了Cerf和Gorny的自组织临界性模型，将平均场相互作用替换为随粒子数变化的一维长程相互作用，发现在高斯情形下极长程相互作用重现平均场行为，而最近邻相互作用则破坏了原有行为。
We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model, which allows us to get precise estimates on boundary connection probabilities. It remains purely discrete, in particular we do not make use of any continuum limit, and it can be used to derive directly several noteworthy results - some new and some not - among which the fact that there is no spontaneous magnetization at criticality, tightness properties for the …
An extension of the Ising-Curie-Weiss model of self-organized criticality with a threshold on the interaction range
本研究扩展了Cerf和Gorny的自组织临界性Ising-Curie-Weiss模型，将平均场相互作用替换为随粒子数变化的一维有限程相互作用，证明自临界行为的相互作用范围阈值为$n^{3/4}$且该阈值尖锐。
We consider the planar Ising model in a finite square box and we replace the temperature parameter with a function depending on the magnetization. This creates a feedback from the spin configuration onto the parameter, which drives the system towards the critical point. Using the finite-size scaling results of arXiv:0811.4507, we show that, when the size of the box grows to infinity, the temperature concentrates around the critical temperature of the planar Ising model on the square lattice.
The purpose of this modest note is to point out that the proof of the recent result of Huchcroft concerning continuity of phase transition in Bernoulli percolation is applicable to the setting of the Ising model with free boundary condition. This observation, together with a recent result of Aizenman, Duminil-Copin, and Sidoravicius implies that the Ising model on amenable quasi-transitive graph with exponential growth undergoes a second order phase transition.
Fluctuating Hydrodynamics of the Ising-Kac-Kawasaki Model and Nonlinear Fluctuations Near Criticality
本研究分析了涨落伊辛-卡克-川崎模型的保守SPDE标度极限，证明其一维重标度方程收敛于随机Cahn-Hilliard方程，解决了相关非线性涨落猜想的简单版本，并证明了小噪声区域的多尺度动力学大偏差及速率函数的Γ收敛。
In this essay, we briefly discuss recent developments, started a decade ago in the seminal work of Smirnov and continued by a number of authors, centered around the conformal invariance of the critical planar Ising model on $\mathbb{Z}^2$ and, more generally, of the critical Z-invariant Ising model on isoradial graphs (rhombic lattices). We also introduce a new class of embeddings of general weighted planar graphs (s-embeddings), which might, in particular, pave the way to true universality results for the planar Ising model.
On the space of Ising configurations on the 2-d square lattice, we consider a family of non Gibbsian measures introduced by using a pair Hamiltonian, depending on an additional inertial parameter $q$. These measures are related to the usual Gibbs measure on $\Z^2$ and turn out to be the marginal of the Gibbs measure of a suitable Ising model on the hexagonal lattice. The inertial parameter $q$ tunes the geometry of the system. The critical behaviour and the decay of correlation functions of these measures are studied thanks to relation with the Random Cluster model.
In this note we overview recent convergence results for correlations in the critical planar nearest-neighbor Ising model. We start with a short discussion of the combinatorics of the model and a definition of fermionic and spinor observables. After that, we illustrate our approach to spin correlations by a derivation of two classical explicit formulae in the infinite-volume limit. Then we describe the convergence results (as the mesh size tends to zero, in arbitrary planar domains) for fermionic correlators, energy-density and spin expectations. Finally, we discuss scaling limits of mixed corr…
We considerably improve upon the recent result of Martinelli and Toninelli on the mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $P$ stochastically dominates the extremal plus phase. An important special case is when $P$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{mix}$ is conjectured to be polynomial in $L$. In [MT] it was shown that for a large enough inverse-temperature $β$ and any $ε>0$ there exists $c=c(β,ε)$ such that $\lim_{L\to\infty}P(T_{mix}\…
Critical window for approximate counting in dense Ising models
研究了临界区域内密集伊辛模型配分函数近似计算的复杂性，证明了在宽度为N^{-1/2+ε}的窗口内近似计算的计算难度，通过全局方法克服了临界点的波动问题。
We try to design a simple model exhibiting self-organized criticality, which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. For a class of symmetric distributions whose density satisfies some integrability conditions, we prove that the sum $S_n$ of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model. The fluctuations are of order $n^{3/4}$, and the limiting law is $C\exp(-λx^4)\,dx$ where $C$ and $λ$ are suitable positive cons…
We consider the sequence of Gibbs measures of Ising models with Kac interaction defined on a periodic two-dimensional discrete torus near criticality. Using the convergence of the Glauber dynamic proven by H. Weber and J.C. Mourrat and a method by H. Weber and P. Tsatsoulis, we show tightness for the sequence of Gibbs measures of the Ising-Kac model near criticality and characterise the law of the limit as the $Φ^4_2$ measure on the torus. Our result is very similar to the one obtained by M. Cassandro, R. Marra and E. Presutti on $\mathbb{Z}^2$, but our strategy takes advantage of the dynamic,…
A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. We define and prove magnetic criticality of a new family of Ising models on planar graphs whose dual is a circle pattern. Our construction includes as a special case the critical isoradial Ising models of Baxter.
It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof of universality and conformal invariance has ever been given, and even physics arguments support (a priori weaker) Möbius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus p…
In a recent and celebrated article, Smirnov [Ann. of Math. (2) 172 (2010) 1435-1467] defines an observable for the self-dual random-cluster model with cluster weight q = 2 on the square lattice $\mathbb{Z}^2$, and uses it to obtain conformal invariance in the scaling limit. We study this observable away from the self-dual point. From this, we obtain a new derivation of the fact that the self-dual and critical points coincide, which implies that the critical inverse temperature of the Ising model equals $1/2\log(1+\sqrt{2})$. Moreover, we relate the correlation length of the model to the large …
We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size $L^2$, inverse temperature $β>\betac$ and overall magnetization conditioned to take the value $\mstar L^2-2\mstar v_L$, where $\betac^{-1}$ is the critical temperature, $\mstar=\mstar(β)$ is the spontaneous magnetization and $v_L$ is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when $v_L^{3/2} L^{-2}$ tends to a definite limit. Specifically, we iden…
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