arXiv subjects

Shangjie Yang

Publications and source records attributed to Shangjie Yang.

7 recordsLinked to original sources

arXiv2024

Cutoff of the simple exclusion process with inhomogeneous conductances

In this paper, we study the mixing time of the simple exclusion process with $k$ particles in the line segment $[1, N]$ with conductances $c^{(N)}(x, x+1)_{1\le x 0$ is the rate of swapping the contents of the two sites $x$ and $ x+1$. Writing $r^{(N)}(x, x+1) := 1/c^{(N)}(x, x+1)$, under the assumption \begin{equation*} \limsup_{N\to \infty}\, \frac{1}{N}\sup_{1< m \le N}\, \left| \sum_{x=2}^m r^{(N)}(x-1, x)- (m-1) \right|\;=\;0\,, \end{equation*} and some further assumptions on $r^{(N)}(x, x+1)_{x \in \mathbb{N} }$ and $k$, we prove that around time $(1+o(1)) (2 π^2)^{-1} N^2 \log k$, the total variation distance to equilibrium of the simple exclusion process drops abruptly from $1$ to $0$.

math.PR↗

arXiv2024

The spectral gap and principle eigenfunction of the random conductance model in a line segment

In this paper, we study the spectral gap and principle eigenfunction of the random walk in the line segment $[1, N]$ with conductances $c^{(N)}(x, x+1)_{1\le x 0$ is the rate of the random walk jumping from site $x$ to site $x+1$ and vice versa. Writing $r^{(N)}(x, x+1) := 1/c^{(N)}(x, x+1)$, under the assumption \begin{equation*} \limsup_{N\to \infty}\, \frac{1}{N}\sup_{1< m \le N}\, \left| \sum_{x=2}^m r^{(N)}(x-1, x)- (m-1) \right|\;=\;0\,, \end{equation*} we prove that the spectral gap, denoted by $\mathrm{gap}_{N}$, of the process satisfies $\mathrm{gap}_{N}=(1+o(1))\pi^2/N^2$ and the principle eigenfunction $g_N$ with $g_N(1)=1$ corresponding to the spectral gap is well approximated by $h_N(x) := \cos\left( (x-1/2)\pi/N \right)$.

math.PR↗

arXiv2023

Typical height of the (2+1)-D Solid-on-Solid surface with pinning above a wall in the delocalized phase

We study the typical height of the (2+1)-dimensional solid-on-solid surface with pinning interacting with an impenetrable wall in the delocalization phase. More precisely, let $Λ_N$ be a $N \times N$ box of $\mathbb{Z}^2$, and we consider a nonnegative integer-valued field $(ϕ(x))_{x \in Λ_N}$ with zero boundary conditions (i.e. $ϕ|_{Λ_N^{\complement}}=0 $) associated with the energy functional $$ \mathcal{V} (ϕ)= β\sum_{x \sim y} \vert ϕ(x)-ϕ(y) \vert- \sum_{x} h \mathbf{1}_{\{ ϕ(x)=0\}},$$ where $β>0$ is the inverse temperature and $h\ge 0$ is the pinning parameter. Lacoin has shown that for sufficiently large $β$, there is a phase transition between delocalization and localization at the critical point $$h_w(β)= \log \left( \frac{e^{4 β}}{e^{4 β}-1}\right).$$ In this paper we show that for $β\ge 1$ and $h \in (0, h_w)$, the values of $ϕ$ concentrate at the height $H= \lfloor (4 β)^{-1} \log N \rfloor$ with constant order fluctuations. Moreover, at criticality $h=h_w$, we provide evidence for the conjectured typical height $H_w= \lfloor (6 β)^{-1} \log N \rfloor$.

math.PR↗

arXiv2023

The branching number of intermediate growth trees

We introduce an "intermediate branching number"(IBN) which captures the branching of intermediate growth trees, similar in spirit to the well-studied branching number of exponential growth trees. We show that the IBN is the critical threshold for several random processes on trees, and analyze the IBN on some examples of interest. Our main result is an algorithm to find spherically symmetric trees with large IBN inside some permutation wreath products. We demonstrate the usefulness of these trees to the study of intermediate growth groups by using them to get the first tight bounds for the firefighter problem on some inetrmediate growth groups.

math.PR↗

arXiv2021

Mixing time for the asymmetric simple exclusion process in a random environment

We consider the simple exclusion process in the integer segment $ [1, N]$ with $k\le N/2$ particles and spatially inhomogenous jumping rates. A particle at site $x\in [ 1, N]$ jumps to site $x-1$ (if $x\ge 2$) at rate $1-ω_x$ and to site $x+1$ (if $x \le N-1$) at rate $ω_x$ if the target site is not occupied. The sequence $ω=(ω_x)_{ x \in \mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\mathbb{E}[ \log ρ_1 ]<0$ where $ρ_1:= (1-ω_1)/ω_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of $N$. More precisely, for the exclusion process with $N^{β+o(1)}$ particles where $β\in [0,1)$, we have in the large $N$ asymptotic $$ N^{\max\left(1,\frac {1}λ, β+ \frac 1 {2λ}\right)+o(1)} \le t_{\mathrm{Mix}}^{N,k} \le N^{C+o(1)}$$ where $λ>0$ is such that $\mathbb{E}[ρ_1^λ]=1$ ($λ=\infty$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $ω$. We conjecture that our lower bound is sharp up to sub-polynomial correction.

math.PR↗

arXiv2020

Metastability for expanding bubbles on a sticky substrate

We study the dynamical behavior of a one dimensional interface interacting with a sticky unpenetrable substrate or wall. The interface is subject to two effects going in opposite directions. Contact between the interface and the substrate are given an energetic bonus while an external force with constant intensity pulls the interface away from the wall. Our interface is modeled by the graph of a one-dimensional nearest-neighbor path on $\mathbb{Z}_+$, starting at $0$ and ending at $0$ after $2N$ steps, the wall corresponding to level-zero the horizontal axis. At equilibrium each path $ξ=(ξ_x)_{x=0}^{2N}$, is given a probability proportional to $λ^{H(ξ)} \exp(\fracσ{N}A(ξ))$, where $H(ξ):=\#\{x \ : ξ_x=0\}$ and $A(ξ)$ is the area enclosed between the path $ξ$ and the $x$-axis. We then consider the classical heat-bath dynamics which equilibrates the value of each $ξ_x$ at a constant rate via corner-flip. Investigating the statics of the model, we derive the full phase diagram in $λ$ and $σ$ of this model, and identify the critical line which separates a localized phase where the pinning force sticks the interface to the wall and a delocalized one, for which the external force stabilizes $ξ$ around a deterministic shape at a macroscopic distance of the wall. On the dynamical side, we identify a second critical line, which separates a rapidly mixing phase (for which the system mixes in polynomial time) to a slow phase where the mixing time grows exponentially. In this slowly mixing regime we obtain a sharp estimate of the mixing time on the $\log$ scale, and provide evidences of a metastable behavior.

math.PR↗

arXiv2019

Cutoff for polymer pinning dynamics in the repulsive phase

We consider the Glauber dynamics for model of polymer interacting with a substrate or wall. The state space is the set of one-dimensional nearest-neighbor paths on $\mathbb{Z}$ with nonnegative integer coordinates, starting at $0$ and coming back to $0$ after $L$ ($L\in 2\mathbb{N}$) steps and the Gibbs weight of a path $ξ=(ξ_x)^{L}_{x=0}$ is given by $λ^{\mathcal{N}(ξ)}$, where $λ\geq 0$ is a parameter which models the intensity of the interaction with the substrate and $\mathcal{N}(ξ)$ is the number of zeros in $ξ$. The dynamics we consider proceeds by updating $ξ_x$ with rate one for each $x=1,\dots, L-1$, in a heat-bath fashion. This model was introduced in [CMT08] with the aim of studying the relaxation to equilibrium of the system. We present new results concerning the total variation mixing time for this dynamics when $λ< 2$, which corresponds to the phase where the effects of the wall's entropic repulsion dominates. For $λ\in [0, 1]$, we prove that the total variation distance to equilibrium drops abruptly from $1$ to $0$ at time $(L^2 \log L)(1+o(1))/π^2$. For $λ\in (1,2)$, we prove that the system also exhibit cutoff at time $(L^2 \log L)(1+o(1))/π^2$ when considering mixing time from "extremal conditions" (that is, either the highest or lowest initial configuration for the natural order on the set of paths). Our results improves both previously proved upper and lower bounds in [CMT08].

math.PR↗