arXiv subjects

Shi-Lei Kong

Publications and source records attributed to Shi-Lei Kong.

5 recordsLinked to original sources

arXiv2024

Cospectral vertices, walk-regular planar graphs and the echolocation problem

We study cospectral vertices on finite graphs in relation to the echolocation problem on Riemannian manifolds. First, We prove a computationally simple criterion to determine whether two vertices are cospectral. Then, we use this criterion in conjunction with a computer search to find minimal examples of various types of graphs on which cospectral but non-similar vertices exist, including minimal walk-regular non-vertex-transitive graphs, which turn out to be non-planar. Moreover, as our main result, we classify all finite 3-connected walk-regular planar graphs, proving that such graphs must be vertex-transitive.

math.CO↗

arXiv2020

Gromov Hyperbolic Graphs Arising From Iterations

For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is Hölder equivalent to the attractor $K$. This setup was taken up to study the probabilistic potential theory on $K$, and the bi-Lipschitz equivalence on $K$. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and Hölder equivalence). We also study a new setup of "weighted" IFS and investigate its connection with the self-similar energy form in the analysis of fractals.

math.MG↗

arXiv2018

Critical exponents of induced Dirichlet forms on self-similar sets

In a previous paper [arXiv:1604.05440], we studied certain random walks on the hyperbolic graphs $X$ associated with the self-similar sets $K$, and showed that the discrete energy ${\mathcal E}_X$ on $X$ has an induced energy form ${\mathcal E}_K$ on $K$ that is a Gagliardo-type integral. The domain of ${\mathcal E}_K$ is a Besov space $Λ^{α, β/2}_{2,2}$ where $α$ is the Hausdorff dimension of $K$ and $β$ is a parameter determined by the "return ratio" of the random walk. In this paper, we study the functional relationship of ${\mathcal E}_X$ and ${\mathcal E}_K$. In particular, we investigate the critical exponents of the $β$ in the domain $Λ^{α, β/2}_{2,2}$ in order for ${\mathcal E}_K$ to be a regular Dirichlet form. We provide some criteria to determine the critical exponents through the effective resistance of the random walk on $X$, and make use of certain electrical network techniques to calculate the exponents for some concrete examples.

math.FA↗

arXiv2018

Random walks and induced Dirichlet forms on compact spaces of homogeneous type

We extend our study of random walks and induced Dirichlet forms on self-similar sets [arXiv:1604.05440, 1612.01708] to compact spaces of homogeneous type $(K, ρ,μ)$. A successive partition on $K$ brings a natural augmented tree structure $(X, E)$ that is Gromov hyperbolic, and the hyperbolic boundary is Hölder equivalent to $K$. We then introduce a class of transient reversible random walks on $(X, E)$ with return ratio $λ$. Using Silverstein's theory of Markov chains, we prove that the random walk induces an energy form on $K$ with $$ {\mathcal E}_K [u] \asymp \iint_{K\times K \setminus Δ} \frac{|u(ξ) - u(η)|^2}{V(ξ, η)ρ(ξ, η)^β} dμ(ξ) dμ(η), $$ where $V(ξ, η)$ is the $μ$-volume of the ball centered at $ξ$ with radius $ρ(ξ, η)$, $Δ$ is the diagonal, and $β$ depends on $λ$. In particular, for an $α$-set in ${\mathbb R}^d$, the kernel of the energy form is of order $\frac{1}{|ξ-η|^{α+β}}$. We also discuss conditions for this energy form to be a non-local regular Dirichlet form.

math.PR↗

arXiv2017

Random walks and induced Dirichlet forms on self-similar sets

Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_HX$ with the Gromov metric is Hölder equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< λ<1$ on $(X, {\mathfrak E})$. We show that the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain precise estimates of the Martin kernel and the Naïm kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate $Θ(ξ, η) \asymp |ξ-η|^{-(α+ β)}$, where $α$ is the Hausdorff dimension of $K$ and $β$ depends on $λ$. For suitable $β$, the kernel defines a regular non-local Dirichlet form on $K$. This extends the results of Kigami concerning random walks on certain trees with Cantor-type sets as boundaries.

math.PR↗