arXiv subjects

Sike Lang

Publications and source records attributed to Sike Lang.

5 recordsLinked to original sources

arXiv2026

A connection inequality for the random-cluster model

For a finite graph $G=(V,E)$, let $ϕ$ be the random-cluster measure with edge parameters $(p_e)_{e\in E}$, cluster weight $q\geq 1$ and free boundary condition. For each $x\in V$ and all nonempty $A,B\subseteq V$, we prove $ϕ(x\leftrightarrow B)\geq ϕ(x\leftrightarrow A)\min_{a\in A}ϕ(a\leftrightarrow B)$. For $q=1$, this is the (slightly generalized) Kozma-Nitzan inequality. Our extension to $q>1$ is based on a recent proof of this inequality by Blanc-Renaudie. As an application, we give a simple proof of sharp upper bounds for one-arm and point-to-plane connection probabilities in the subcritical regime.

math.PR↗

arXiv2026

Phase transition in the antiferromagnetic Ising model with external field

Consider the nearest-neighbor antiferromagnetic Ising model on $\mathbb{Z}^d$, $d\ge2$, at inverse temperature $β$ and in a uniform external field $h$. Let $β_c(d)$ be the critical inverse temperature of the zero-field Ising model. We prove that for every $β>β_c(d)$, the Gibbs state is not unique whenever $|h|$ is sufficiently small. This extends a previous result of Dobrushin (1968) from sufficiently large $β$ to every $β>β_c(d)$.

math.PR↗

arXiv2026

Coexistence of infinite clusters for percolation and Ising model on $\mathbb{Z}^d$

For independent bond percolation on $\mathbb{Z}^d$ with parameter $p$, let $p_c^b(d)$ be the critical probability. We prove that for each $d \geq 9$, there is $ε_d>0$ such that for each $p \in (p_c^b(d), p_c^b(d)+ε_d)$, the complement of the infinite open cluster stochastically dominates a supercritical site percolation on $\mathbb{Z}^d$. This improves the previous results by Grimmett, Holroyd and Kozma 2014, and Bock, Damron, Newman and Sidoravicius 2020. Numerical estimates of $p_c^b(d)$ and $p_c^s(d)$ (the site critical probability) suggest that a similar stochastic domination result holds for all $d \geq 4$. For the Ising model on $\mathbb{Z}^d$ with inverse temperature $β$, let $β_c(d)$ be the critical inverse temperature. We prove that for each $d \geq 8$, there is $ε_d>0$ such that for each $β\in [0,β_c(d)+ε_d)$, the $+$ spins under the minus phase stochastically dominate a supercritical site percolation on $\mathbb{Z}^d$. This improves the previous result of Aizenman, Bricmont and Lebowitz 1987. Numerical estimates of $β_c(d)$ and $p_c^s(d)$ suggest that a similar stochastic domination result holds for all $d \geq 5$.

math.PR↗

arXiv2026

The free state for the Potts model on Cayley trees is either extremal or glassy

For the Potts model on the Cayley tree $\mathbb{T}^d$ with branching factor $d\geq 2$, we consider the free state which is obtained as the limiting Gibbs measure with free boundary conditions. We prove that the free state is either extremal or glassy (i.e., whose decomposition into extremal Gibbs measures contains uncountably many components). As a corollary, the free state for the Ising model on $\mathbb{T}^d$ is glassy if and only if the inverse temperature $β>\operatorname{arctanh}(1/\sqrt{d})$; this generalizes a previous result by Gandolfo, Maes, Ruiz and Shlosman (2020) from a very low temperature regime to the entire spin-glass regime.

math.PR↗

arXiv2025

Percolation of both signs in a triangular-type 3D Ising model above $T_c$

Let $\mathbb{T}$ be the two-dimensional triangular lattice, and $\mathbb{Z}$ the one-dimensional integer lattice. Let $\mathbb{T}\times \mathbb{Z}$ denote the Cartesian product graph. Consider the Ising model defined on this graph with inverse temperature $β$ and external field $h$, and let $β_c$ be the critical inverse temperature when $h=0$. We prove that for each $β\in[0,β_c)$, there exists $h_c(β)>0$ such that both a unique infinite $+$cluster and a unique infinite $-$cluster coexist whenever $|h|<h_c(β)$. The same coexistence result also holds for the three-dimensional triangular lattice.

math.PR↗