Kyuhyeon Choi | Searcharxiv

arXiv2026

Dimension-decaying diffusion processes as the scaling limit of condensing zero-range processes

In this article, we prove that, on the diffusive time scale, condensing zero-range processes converge to a dimension-decaying diffusion process on the simplex \[ Σ= \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in S} x_i = 1\}, \] where $S$ is a finite set. This limiting diffusion has the distinctive feature of being absorbed at the boundary of the simplex. More precisely, once the process reaches a face \[ Σ_A = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in A} x_i = 1\}, \qquad A \subset S, \] it remains confined to this set and evolves in the corresponding lower-dimensional simplex according to a new diffusion whose parameters depend on the subset $A$. This mechanism repeats itself, leading to successive reductions of the dimension, until one of the vertices of the simplex is reached in finite time. At that point, the process becomes permanently trapped. The proof relies on a method to extend the domain of the associated martingale problem, which may be of independent interest and useful in other contexts.

math.PR↗

arXiv2024

A $Γ$-convergence of level-two large deviation for metastable systems: The case of zero-range processes

This study explores the relationship between the precise asymptotics of the level-two large deviation rate function and the behavior of metastable stochastic systems. Initially identified for overdamped Langevin dynamics (Ges{ù} et al., SIAM J Math Anal 49(4), 3048-3072, 2017), this connection has been validated across various models, including random walks in a potential field. We extend this connection to condensing zero-range processes, a complex interacting particle system. Specifically, we investigate a certain class of zero-range processes on a fixed graph $G$ with $N > 0$ particles and interaction parameter $α> 1$. On the time scale $N^2$, this process behaves like an absorbing-type diffusion and converges to a condensed state where all particles occupy a single vertex of $G$ as $N$ approaches infinity. Once condensed, on the time scale $N^{1+α}$, the condensed site moves according to a Markov chain on $G$, showing metastable behavior among condensed states. The time scales $N^2$ and $N^{1+α}$ are called the pre-metastable and metastable time scales. It is conjectured that this behavior is encapsulated in the level-two large deviation rate function $\mathcal{I}_N$ of the zero-range process. Specifically, it is expected that the $Γ$-expansion of $\mathcal{I}_N$ can be expressed as:$$\mathcal{I}_N = \frac{1}{N^2} \mathcal{K} + \frac{1}{N^{1+α}} \mathcal{J},$$ where $\mathcal{K}$ and $\mathcal{J}$ are the level-two large deviation rate functions of the absorbing diffusion processes and the Markov chain on $G$. We rigorously prove this $Γ$-expansion by developing a methodology for $Γ$-convergence in the pre-metastable time scale and establishing a link between the resolvent approach to metastability (Landim et al., J Eur Math Soc, 2023. arXiv:2102.00998) and the $Γ$-expansion in the metastable time scale.

math.PR↗

arXiv2020

A Central Limit Theorem for Rosen Continued Fractions

We prove a central limit theorem for Birkhoff sums of the Rosen continued fraction algorithm. A Lasota-Yorke bound is obtained for general one-dimensional continued fractions with the bounded variation space, which implies quasi-compactness of the transfer operator. The main result is a direct proof of the existence of a spectral gap, assuming a certain behavior of the transformation when iterated. This condition is explicitly proved for the Rosen system. We conclude via well-known results of A. Broise that the central limit theorem holds.

math.DS↗