arXiv subjects

Aoteng Xia

Publications and source records attributed to Aoteng Xia.

4 recordsLinked to original sources

arXiv2026

Scaling limits of the single-curve interface and outermost loops in the planar random field Ising model

We prove that the interface separating $+1$ and $-1$ spins in the near-critical planar random field Ising model (RFIM) with Dobrushin boundary conditions has a scaling limit, whose law is conformally covariant and almost surely absolutely continuous with respect to SLE$_3$. The limiting curve can be seen as a massive version of SLE$_3$ in the sense of Makarov and Smirnov, but in a random environment. We then show that the outermost spin loops of the near-critical planar RFIM with $+1$ boundary conditions have subsequential limits and that any of these limits is almost surely singular with respect to CLE$_3$. This dichotomy between absolute continuity of the single interface and singularity of the outermost loops reflects the fact that a single interface does not explore enough of the magnetization field of the near-critical RFIM to detect the singularity of this field with respect to the critical Ising magnetization field, whereas the outermost spin loops do.

math.PR↗

arXiv2025

A phase transition for the two-dimensional random field Ising/FK-Ising model

We study the total variation (TV) distance between the laws of the 2D Ising/FK-Ising model in a box of side-length $N$ with and without an i.i.d.\ Gaussian external field with variance $ε^2$. Letting the external field strength $ε= ε(N)$ depend on the size of the box, we derive a phase transition for each model depending on the order of $ε(N)$. For the random field Ising model, the critical order for $ε$ is $N^{-1}$. For the random field FK-Ising model, the critical order depends on the temperature regime: for $T>T_c$, $T=T_c$ and $T\in (0, T_c)$ the critical order for $ε$ is, respectively, $N^{-\frac{1}{2}}$, $N^{-\frac{15}{16}}$ and $N^{-1}$. In each case, as $N \to \infty$ the TV distance under consideration converges to $1$ when $ε$ is above the respective critical order and converges to $0$ when below.

math.PR↗

arXiv2024

A phase transition and critical phenomenon for the two-dimensional random field Ising model

We study the random field Ising model in a two-dimensional box with side length $N$ where the external field is given by independent normal variables with mean $0$ and variance $ε^2$. Our primary result is the following phase transition at $T = T_c$: for $ε\ll N^{-7/8}$ the boundary influence (i.e., the difference between the spin averages at the center of the box with the plus and the minus boundary conditions) decays as $N^{-1/8}$ and thus the disorder essentially has no effect on the boundary influence; for $ε\gg N^{-7/8}$, the boundary influence decays as $N^{-\frac{1}{8}}e^{-Θ(ε^{8/7}\, N)}$ (i.e., the disorder contributes a factor of $e^{-Θ(ε^{8/7}\, N)}$ to the decay rate). For a natural notion of the correlation length, i.e., the minimal size of the box where the boundary influence shrinks by a factor of $2$ from that with no external field, we also prove the following: as $ε\downarrow 0$ the correlation length transits from $Θ(ε^{-8/7})$ at $T_c$ to $e^{Θ(ε^{-4/3}\,\,)}$ for $T < T_c$.

math.PR↗

arXiv2022

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

For $d\geq 3$, we study the Ising model on $\mathbb Z^d$ with random field given by $\{εh_v: v\in \mathbb Z^d\}$ where $h_v$'s are independent normal variables with mean 0 and variance 1. We show that for any $T < T_c$ (here $T_c$ is the critical temperature without disorder), long range order exists as long as $ε$ is sufficiently small depending on $T$. Our work extends previous results of Imbrie (1985) and Bricmont--Kupiainen (1988) from the very low temperature regime to the entire low temperature regime.

math.PR↗