arXiv subjects

Junchi Zuo

Publications and source records attributed to Junchi Zuo.

5 recordsLinked to original sources

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 $\epsilon$-convergence time $\tau_\epsilon$ as the first time when the maximum and the minimum of $f_t$ differ by at most $\epsilon$. It is known that if the initial opinions $(f_0(v))_{v \in V}$ are bounded in $L^\infty$, then $\mathbb{E}(\tau_\epsilon)$ is at most $C_\epsilon \log^2 n$ for $\epsilon \in (0, 1]$. We assume instead that the $L^p$ norm of $f_0(v)$ is at most $1$ for every $v \in V$. For fixed $\epsilon \in (0, 1]$, and show that $\mathbb{E}(\tau_\epsilon) = \widetilde{O}(n^{\beta_p})$ up to logarithmic terms, where $\beta_p := \max(3 - p, 2/p)$. Moreover, this power law is tight on cycle graphs.

math.PR↗

arXiv2025

How reactive gambling can backfire: ruin probability is increasing in $p$, H\"older continuous in initial fortune

A gambler with an initial fortune $x$ starts by betting a dollar, then doubles the bet after every win and halves the bet after every loss. Let $p\in (0,1)$ be the probability of winning for each round. We show that the gambler survives with positive probability if and only if $p < 1/2$ and $x > 2$. Moreover, the ruin probability is increasing and real-analytic in $p$, but a singular, H\"older continuous function of $x$.

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 $\lVert f_t - h \rVert_{\infty} \le \epsilon$ by $\tau_p(\epsilon).$ 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 $\E[\tau_p(\epsilon)]$ is at most $n^{\beta_p}$ (up to logarithmic factors in $\frac{n}{\epsilon}$ for $p \in [2, \infty)$, and polynomial factors in $\epsilon^{-1}$ for $p \in (1, 2)$), where $\beta_p=\max\big(\frac{2p}{p-1},3\big)$. This matches the definition of $\beta_p$ given in \cite{noboundarycase} and answers Question 6.2 in that paper. The exponent $\beta_p$ 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↗