树上局域Gibbs测度与去局域梯度Gibbs测度的共存

Phase coexistence of gradient Gibbs states

We consider the (scalar) gradient fields $η=(η_b)$--with $b$ denoting the nearest-neighbor edges in $\Z^2$--that are distributed according to the Gibbs measure proportional to $\texte^{-βH(η)}ν(\textdη)$. Here $H=\sum_bV(η_b)$ is the Hamiltonian, $V$ is a symmetric potential, $β>0$ is the inverse temperature, and $ν$ is the Lebesgue measure on the linear space defined by imposing the loop condition $η_{b_1}+η_{b_2}=η_{b_3}+η_{b_4}$ for each plaquette $(b_1,b_2,b_3,b_4)$ in $\Z^2$. For convex $V$, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their…

作为可恢复系统的最大硬核模型：吉布斯测度与相共存

The maximal hard-core model as a recoverable system: Gibbs measures and phase coexistence

该研究将最大硬核模型作为可恢复系统，引入格点局部恢复区域相互作用势，证明高低温下吉布斯测度的相关性质，并通过围道模型与Pirogov-Sinai理论验证高低活性下的相共存。

Phase Coexistence for the Hard-Core Model on ${\mathbb Z}^2$

The hard-core model has attracted much attention across several disciplines, representing lattice gases in statistical physics and independent sets in discrete mathematics and computer science. On finite graphs, we are given a parameter $λ$, and an independent set $I$ arises with probability proportional to $λ^{|I|}$. On infinite graphs a Gibbs measure is defined as a suitable limit with the correct conditional probabilities, and we are interested in determining when this limit is unique and when there is phase coexistence, i.e., existence of multiple Gibbs measures. It has long been conjectur…

A proof of the Gibbs-Thomson formula in the droplet formation regime

We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic derivation of the Gibbs-Thomson formula for the deviation of the pressure and the density away from their equilibrium values which, according to the interpretation of the classical thermodynamics, appears due to the presence of a curved interface. The general--albeit heuristic--reasoning is corroborated by a rigorous proof in the case of the two-dimensional Ising lattice gas.

正则树上非平移不变的梯度Gibbs测度的存在性

Existence of gradient Gibbs measures on regular trees which are not translation invariant

该研究在仅假设转移算子可和的条件下，通过两层隐马尔可夫模型与内部q自旋模型构造，证明了正则树上Z值自旋模型存在非平移不变的离域梯度Gibbs测度，证明基于大q下内部模型离散动力系统的局部伪不稳定流形性质。

Phase transition for the Ising model on the Critical Lorentzian triangulation

Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of Peierls method. The critical temperature is shown to be constant a.s.