相关论文: Lamplighters, Diestel-Leader graphs, random walks,…
The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size -3,-5,-7,... . For such paths, we find the generating functions of them, according to length, ending at level $i$,…
We study fine structure related to finitely supported random walks on infinite finitely generated discrete groups, largely motivated by dimension group techniques. The unfaithful extreme harmonic functions (defined only on proper space-time…
In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We…
Two graphs are isomorphic exactly when they admit the same number of homomorphisms from every graph. Hence, a graph is recognized up to isomorphism by homomorphism counts over the class of all graphs. Restricting to a specific graph class…
We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon $-energy functional with Dirichlet values and we establish the $ \Gamma $-limit. The minimizers of the limiting functional are closely related to minimizing…
We exhibit finitely generated groups with prescribed Poincar\'e profiles. It can be prescribed for functions between $n/\log n$ and linear, and is sharp for functions at least $n/(\log\log n)$. Those profiles were introduced by Hume, Mackay…
We compute the cone types of the Cayley graph of the modular group $\mathrm{PSL}(2,\mathbf{Z})$ associated with the standard system of generators ${\small\left\{\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0…
On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We…
The Dunkl Laplacian is used to define the Hamiltonian of a modified quantum harmonic oscillator, associated with any finite reflection group. The potential is a sum of the inverse squares of the linear functions whose zero sets are the…
The coherent configuration $\mathsf{WL}(X)$ of a graph $X$ is the smallest coherent configuration on the vertices of $X$ that contains the edge set of $X$ as a relation. The aim of the paper is to study $\mathsf{WL}(X)$ when $X$ is a…
We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a…
The field- and temperature-dependent de Haas-van Alphen oscillations spectrum is studied for an ideal two-dimensional compensated metal whose Fermi surface is made of a linear chain of successive orbits with electron and hole character,…
In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the…
A classical construction associates to a transient random walk on a discrete group $\Gamma$ a compact $\Gamma$-space $\partial_M \Gamma$ known as the Martin boundary. The resulting crossed product $C^*$-algebra $C(\partial_M \Gamma)…
Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on…
For a graph $G = (V, E)$, the $\gamma$-graph of $G$, denoted $G(\gamma) = (V(\gamma), E(\gamma))$, is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent in…
Regular ring lattices (RRLs) are defined as peculiar undirected circulant graphs constructed from a cycle graph, wherein each node is connected to pairs of neighbors that are spaced progressively in terms of vertex degree. This kind of…
The WL-rank of a graph $\Gamma$ is defined to be the rank of the coherent configuration of $\Gamma$. The WL-dimension of $\Gamma$ is defined to be the smallest positive integer $m$ for which $\Gamma$ is identified by the $m$-dimensional…
Stochastic treatments of magnetic resonance spectroscopy and optical spectroscopy require evaluations of functions like <exp(i int_0^t Q(s)ds)>, where t is time, Q(s) is the value of a stochastic process at time s, and the angular brackets…
In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem…