arXiv subjects

Junchi Zuo

Publications and source records attributed to Junchi Zuo.

5 recordsLinked to original sources

arXiv2026

Survival probability in reactive gambling: monotonicity, singularity, and Hölder continuity

A gambler starts with fortune $x$ and unit bet, then aims to double the bet after every win and halve it after every loss, while the actual bet is capped by the current fortune. Each round is won independently with probability $p\in(0,1)$. We study $f(x,p)$, the probability that the gambler is never ruined. We prove that $f(x,p)>0$ exactly when $x>2$ and $p<1/2$. For fixed $x>2$, $f(x,\cdot)$ is strictly decreasing and real analytic on $(0,1/2)$, and vanishes on $[1/2,1)$: playing a more favourable game lowers the chance of surviving it. For fixed $p\in(0,1/2)$, $f(\cdot,p)$ is strictly increasing and singular continuous on $[2,\infty)$. An elementary two-step argument gives an explicit global Hölder exponent, with constant one, which is strictly larger than $\log_2(1+p)$ and tends to $\frac12\log_2(111/46)=0.635426\ldots$ as $p\uparrow1/2$; we place it in a hierarchy of computable exponents and show, by an exact finite computation, that the optimal exponent is strictly smaller than the pointwise exponent at $x=2$ once $p$ is close to $1/2$. We determine that boundary exponent, and give the complete phase diagram when wins and losses use unequal multipliers.

math.PR↗

arXiv2026

Edge-averaging dynamics on finite graphs: moment dependence

We study the edge-averaging process on a finite, connected graph $G = (V, E)$. Initially, the vertices in $V$ are endowed with i.i.d.\ real-valued opinions $(f_0(v))_{v \in V}$. Edges are activated according to i.i.d.\ Poisson clocks of rate $1$; when an edge is activated, the opinions at its endpoints are replaced by their average. Let $f_t(v)$ denote the opinion at $v$ at time $t$.Define the $ε$-convergence time $τ_ε$ as the first time when the maximum and the minimum of $f_t$ differ by at most $ε$. It is known that if the initial opinions $(f_0(v))_{v \in V}$ are bounded in $L^\infty$, then $\mathbb{E}(τ_ε)$ is at most $C_ε\log^2 n$ for $ε\in (0, 1]$. We assume instead that the $L^p$ norm of $f_0(v)$ is at most $1$ for every $v \in V$. For fixed $ε\in (0, 1]$, and show that $\mathbb{E}(τ_ε) = \widetilde{O}(n^{β_p})$ up to logarithmic terms, where $β_p := \max(3 - p, 2/p)$. Moreover, this power law is tight on cycle graphs.

math.PR↗

arXiv2025

Convergence rate of $\ell^p$-relaxation on a graph to a $p$-harmonic function with given boundary values

We analyze the following dynamics on a connected graph $(V,E)$ with $n$ vertices. Let $V = I \bigcup B$, where the set of interior vertices $I \ne \emptyset$ is disjoint from the set of boundary vertices $B \neq \emptyset$. Given $p > 1$ and an initial opinion profile $f_0: V \to [0,1]$, at each integer step $t \ge 1$ a uniformly random vertex $v_t \in I$ is selected, and the opinion there is updated to the value $f_{t}(v_t)$ that minimizes the sum $\sum_{w \sim v_t} \lvert f_t(v_t)-f_{t-1}(w) \rvert^p$ over neighbours $w$ of $v_t$. The case $p=2$ yields linear averaging dynamics, but for all $p \ne 2$ the dynamics are nonlinear. It is well known that almost surely, $f_t$ converges to the $p$-harmonic extension $h$ of $f_0 \vert_{B}$. Denote the number of steps needed to obtain $\lVert f_t - h \rVert_{\infty} \le ε$ by $τ_p(ε).$ Recently, Amir, Nazarov, and Peres~\cite{noboundarycase} analyzed the same dynamics without boundary. For individual graphs, adding boundary values can slow down the convergence considerably; indeed, when $p = 2$ the approximation time is controlled by the hitting time of the boundary by random walk, and hitting times can be much larger than mixing times, which control the convergence when $B=\emptyset$. Nevertheless, we show that for all graphs with $n$ vertices, the mean approximation time $\E[τ_p(ε)]$ is at most $n^{β_p}$ (up to logarithmic factors in $\frac{n}ε$ for $p \in [2, \infty)$, and polynomial factors in $ε^{-1}$ for $p \in (1, 2)$), where $β_p=\max\big(\frac{2p}{p-1},3\big)$. This matches the definition of $β_p$ given in \cite{noboundarycase} and answers Question 6.2 in that paper. The exponent $β_p$ is optimal in both settings. We also prove sharp bounds for $n$-vertex graphs with given average degree, that are technically more challenging.

math.PR↗

arXiv2022

Successive vertex orderings of fully regular graphs

A graph G = (V,E) is called fully regular if for every independent set $I\subset V$ , the number of vertices in $V\setminus$ I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph. As an application of our results, we give alternative proofs of two theorems of Stanley and Gao + Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph. As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.

math.CO↗

arXiv2019

A graph inequality on the common neighbourhood

In this note we prove a graph inequality based on the sizes of the common neighbourhoods. We also characterize the extremal graphs that achieve the equality. The result was first discovered as a consequence of the classical Forster's theorem in electric networks. We also present a short combinatorial proof that was inspired by a similar inequality related to the celebrated Turán's theorem.

math.CO↗