arXiv subjects

Chang-Long Yao

Publications and source records attributed to Chang-Long Yao.

9 recordsLinked to original sources

arXiv2023

Limit of the Wulff crystal when approaching criticality for isoperimetry in 2D percolation

We consider isoperimetric sets, i.e., sets with minimal vertex boundary for a prescribed volume, of the infinite cluster of supercritical site percolation on the triangular lattice. Let $p$ be the percolation parameter and let $p_c$ be the critical point. By adapting the proof of Biskup, Louidor, Procaccia and Rosenthal [6] for isoperimetry in bond percolation on the square lattice, we show that the isoperimetric sets, when suitably rescaled, converge almost surely to a translation of the normalized Wulff crystal $\widehat{W}_p$. More importantly, we prove that $\widehat{W}_p$ tends to a Euclidean disk as $p\downarrow p_c$. This settles the site version of a conjecture proposed in [6]. A key input to the proof is the convergence of the limit shapes for near-critical Bernoulli first-passage percolation proved by the author recently.

math.PR↗

arXiv2022

Convergence of limit shapes for 2D near-critical first-passage percolation

We consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. For each $p\in(0,p_c)$, let $\mathcal {B}(p)$ be the limit shape in the classical "shape theorem", and let $L(p)$ be the correlation length. We show that as $p\uparrow p_c$, the rescaled limit shape $L(p)^{-1}\mathcal {B}(p)$ converges to a Euclidean disk. This improves a result of Chayes et al. [J. Stat. Phys. 45 (1986) 933--951]. The proof relies on the scaling limit of near-critical percolation established by Garban et al. [J. Eur. Math. Soc. 20 (2018) 1195--1268], and uses the construction of the collection of continuum clusters in the scaling limit introduced by Camia et al. [Springer Proceedings in Mathematics \& Statistics, 299 (2019) 44--89].

math.PR↗

arXiv2018

Asymptotics for 2D critical and near-critical first-passage percolation

We study Bernoulli first-passage percolation (FPP) on the triangular lattice $\mathbb{T}$ in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. Denote by $\mathcal {C}_{\infty}$ the infinite cluster with 0-time sites when $p>p_c$, where $p_c=1/2$ is the critical probability. Denote by $T(0,\mathcal {C}_{\infty})$ the passage time from the origin 0 to $\mathcal {C}_{\infty}$. First we obtain explicit limit theorem for $T(0,\mathcal {C}_{\infty})$ as $p\searrow p_c$. The proof relies on the limit theorem in the critical case, the critical exponent for correlation length and Kesten's scaling relations. Next, for the usual point-to-point passage time $a_{0,n}$ in the critical case, we construct subsequences of sites with different growth rate along the axis. The main tool involves the large deviation estimates on the nesting of CLE$_6$ loops derived by Miller, Watson and Wilson (2016). Finally, we apply the limit theorem for critical Bernoulli FPP to a random graph called cluster graph, obtaining explicit strong law of large numbers for graph distance.

math.PR↗

arXiv2018

Critical first-passage percolation starting on the boundary

We consider first-passage percolation on the two-dimensional triangular lattice $\mathcal{T}$. Each site $v\in\mathcal{T}$ is assigned independently a passage time of either $0$ or $1$ with probability $1/2$. Denote by $B^+(0,n)$ the upper half-disk with radius $n$ centered at $0$, and by $c_n^+$ the first-passage time in $B^+(0,n)$ from $0$ to the half-circular boundary of $B^+(0,n)$. We prove \[\lim_{n\rightarrow\infty}\frac{c_n^+}{\log n}=\frac{\sqrt{3}}{2π}~ a.s.,~\lim_{n\rightarrow\infty}\frac{E c_n^+}{\log n}=\frac{\sqrt{3}}{2π},~\lim_{n\rightarrow\infty}\frac{\mathrm{Var}(c_n^+)}{\log n}=\frac{2\sqrt{3}}π-\frac{9}{π^2}.\] These results enable us to prove limit theorems with explicit constants for any first-passage time between boundary points of Jordan domains. In particular, we find the explicit limit theorems for the cylinder point to point and cylinder point to line first-passage times.

math.PR↗

arXiv2017

Multi-arm incipient infinite clusters in 2D: scaling limits and winding numbers

We study the alternating $k$-arm incipient infinite cluster (IIC) of site percolation on the triangular lattice $\mathbb{T}$. Using Camia and Newman's result that the scaling limit of critical site percolation on $\mathbb{T}$ is CLE$_6$, we prove the existence of the scaling limit of the $k$-arm IIC for $k=1,2,4$. Conditioned on the event that there are open and closed arms connecting the origin to $\partial \mathbb{D}_R$, we show that the winding number variance of the arms is $(3/2+o(1))\log R$ as $R\rightarrow \infty$, which confirms a prediction of Wieland and Wilson (2003). Our proof uses two-sided radial SLE$_6$ and coupling argument. Using this result we get an explicit form for the CLT of the winding numbers, and get analogous result for the 2-arm IIC, thus improving our earlier result.

math.PR↗

arXiv2016

Limit theorems for critical first-passage percolation on the triangular lattice

Consider (independent) first-passage percolation on the sites of the triangular lattice $\mathbb{T}$. Denote the passage time of the site $v$ in $\mathbb{T}$ by $t(v)$, and assume that $P(t(v)=0)=P(t(v)=1)=1/2$. Denote by $b_{0,n}$ the passage time from 0 to the halfplane $\{v\in\mathbb{T}:\mbox{Re}(v)\geq n\}$, and by $T(0,nu)$ the passage time from 0 to the nearest site to $nu$, where $|u|=1$. We prove that as $n\rightarrow\infty$, $b_{0,n}/\log n\rightarrow 1/(2\sqrt{3}π)$ a.s., $E[b_{0,n}]/\log n\rightarrow 1/(2\sqrt{3}π)$ and Var$[b_{0,n}]/\log n\rightarrow 2/(3\sqrt{3}π)-1/(2π^2)$; $T(0,nu)/\log n\rightarrow 1/(\sqrt{3}π)$ in probability but not a.s., $E[T(0,nu)]/\log n\rightarrow 1/(\sqrt{3}π)$ and Var$[T(0,nu)]/\log n\rightarrow 4/(3\sqrt{3}π)-1/π^2$. This answers a question of Kesten and Zhang (1997) and improves our previous work (2014). From this result, we derive an explicit form of the central limit theorem for $b_{0,n}$ and $T(0,nu)$. A key ingredient for the proof is the moment generating function of the conformal radii for conformal loop ensemble CLE$_6$, given by Schramm, Sheffield and Wilson (2009).

math.PR↗

arXiv2014

Law of large numbers for critical first-passage percolation on the triangular lattice

We study the site version of (independent) first-passage percolation on the triangular lattice $\mathbb{T}$. Denote the passage time of the site $v$ in $\mathbb{T}$ by $t(v)$, and assume that $P(t(v)=0)=P(t(v)=1)=1/2$. Denote by $a_{0,n}$ the passage time from $\textbf{0}$ to $(n,0)$, and by $b_{0,n}$ the passage time from $\textbf{0}$ to the halfplane $\{(x,y):x\geq n\}$. We prove that there exists a constant $0<μ<\infty$ such that as $n\rightarrow\infty$, $a_{0,n}/\log n\rightarrow μ$ in probability and $b_{0,n}/\log n\rightarrow μ/2$ almost surely. This result confirms a prediction of Kesten and Zhang (Probab. Theory Relat. Fields \textbf{107}: 137--160, 1997). The proof relies on the existence of the full scaling limit of critical site percolation on $\mathbb{T}$, established by Camia and Newman.

math.PR↗

arXiv2013

The asymptotic size of the largest component in random geometric graphs with some applications

For the size of the largest component in a supercritical random geometric graph, this paper estimates its expectation which tends to a polynomial on a rate of exponential decay, and sharpens its asymptotic result with a central limit theory. Similar results can be obtained for the size of biggest open cluster, and for the number of open clusters of percolation on a box, and so on.

math.PR↗

arXiv2013

A CLT for winding angles of the arms for critical planar percolation

Consider critical percolation in two dimensions. Under the condition that there are k disjoint alternating black and white arms crossing the annulus A(l,n), we prove a central limit theorem and variance estimates for the winding angles of the arms (as n\rightarrow \infty, l fixed). This result confirms a prediction of Beffara and Nolin (Ann. Probab. 39: 1286--1304, 2011). Using this theorem, we also get a CLT for the multiple-armed incipient infinite cluster (IIC) measures.

math.PR↗