arXiv subjects

Wolfgang Woess

Publications and source records attributed to Wolfgang Woess.

At least 19 recordsLinked to original sources

arXiv2026

Graphs of group actions and group actions on trees

Bass-Serre theory provides a powerful framework for studying group actions on trees. While extremely effective for structural questions in group theory, it is less suited to the systematic construction of group actions with prescribed local behaviour. Motivated by local-to-global constructions such as the Burger-Mozes universal groups and local action diagrams, we develop an analogue of Bass-Serre theory for group actions. The central object of study in our are graphs of group actions, combinatorial structures similar to graphs of groups from Bass-Serre theory, encoding compatible local permutation actions on a base graph. From these we can construct groups which act on tree-like graphs called scaffoldings and hence also on trees. We prove uniqueness and universality results for the resulting groups and show that our framework unifies and generalises (among other known constructions) both graphs of groups and local action diagrams. Remarkably, we are able to encapsulate the full generality of the former while still allowing for efficient construction of groups with certain local properties like in the latter.

math.GR↗

arXiv2026

Notes on hyperbolic branching Brownian motion

Euclidean branching Brownian motion (BBM) has been intensively studied during many decades by renowned researchers. BBM on hyperbolic space has received less attention. A profound study of Lalley and Sellke (1997) provided insight on the recurrent, resp. transient regimes of BBM on the Poincare' disk. In particular, they determined the Hausdorff dimension of the limit set on the boundary circle in dependence on the fission rate of the branching particles. In the present notes, further features are exhibited. The rates of the maximal and minimal hyperbolic distances to the starting point are determined, as well as refined asymptotic estimates in the transient regime. The other main issues studied here concern the behaviour of the empiricial distributions of the branching population, as time goes to infinity, and their convergence to an infinitely supported random limit probability measure on the boundary.

math.PR↗

arXiv2026

Some old and basic facts about random walks on groups

This note contains old instead of new results about random walks on groups, which may serve as a small supplement to the author's monograph ``Random Walks on Infinite Graphs and Groups'' (Cambridge Univ. Press 2000/2009). First, we exhibit a basic exercise on the periodicity classes of random walk. The second topic concerns some basics on ratio limits for random walks, which had been published ``only'' in German in the 1970ies.

math.PR↗

arXiv2026

Diffusion on homogeneous ultrametric spaces: the contributions of Alessandro Figà-Talamanca

Alessandro Figà-Talamanca (1938-2023) was an influential Italian mathematician, scientific leader of the Italian group of harmonic analysis for many years. Since the late 1970ies, his interest focussed on harmonic analysis on free groups and trees. In the later years of his scientific work he became also interested in diffusion processes on homogeneous ultrametric spaces such as local fields and totally disconnected Abelian groups. This is related with the close connection of those spaces with trees and their boundaries and concerns, in particular, the construction of such processes via discrete-time walks on trees. The present notes provide rather detailed comments on this part of his work and the related, quite abundant literature. This is intended to become part of a volume of selected papers by Figà-Talamanca, accompanied by comments such as the present text.

math.PR↗

arXiv2025

The spectra of graph substitutions

Let $(X,E_X)$ and $(V,E_V)$ be finite connected graphs without loops. We assume that $V$ has two distinguished vertices $a,b$ and an automorphism $γ$ which exchanges $a$ and~$b$. The $V$-edge substitution of $X$ is the graph $X[V]$ where each edge $[x,y] \in E_X$ is replaced by a copy of $V$, identifying $x$ with $a$ and $y$ with $b$ or vice versa. (The latter choice does not matter; it yields isomorphic graphs.) The aim is to describe the spectrum of $X[V]$ in terms of the spectra of $X$ and $V$. Instead of the spectra of the adjacency matrices, we consider the versions which are normalised by dividing each row by the row sum (the vertex degree). These are stochastic, reversible matrices, and our approach applies more generally to reversible transition matrices corresponding to arbitrary positive edge weights invariant under $γ$. We write $P$ for the transition matrix over $X$ and $Q$ for the one over $V$. Together, they induce the matrix $P_*$ over $X[V]$. The main part of the spectrum of $P_*$ is the response of the natural frequencies of $X$ to substituting $V$, given by a functional equation coming from a rational function induced by~$Q$. A second part comes from specific eigenvalues of $Q$, if present. Finally, there is the part of $\mathsf{spec}(P_*)$ whose eigenfunctions have $X$ as a nodal set. The results depend on issues like whether $X$ has circles of even length and on the eigenvalues of the restriction of $Q$ to $V \setminus \{ a,b\}$, which are classified into 4 possible types. Quite subtle is the issue of determining the multiplicities of the latter as eigenvalues of $P_*$ in terms of the input.

math.CO↗

arXiv2023

Polyharmonic potential theory on the Poincaré disk

We consider the open unit disk $\mathbb{D}$ equipped with the hyperbolic metric and the associated hyperbolic Laplacian $\mathfrak{L}$. For $λ\in \mathbb{C}$ and $n \in \mathbb{N}$, a $λ$-polyharmonic function of order $n$ is a function $f: \mathbb{D} \to \mathbb{C}$ such that $(\mathfrak{L}- λ\, I)^n f = 0$. If $n =1$, one gets $λ$-harmonic functions. Based on a Theorem of Helgason on the latter functions, we prove a boundary integral representation theorem for $λ$-polyharmonic functions. For this purpose, we first determine $n^{\text{th}}$-order $λ$-Poisson kernels. Subsequently, we introduce the $λ$-polyspherical functions and determine their asymptotics at the boundary $\partial \mathbb{D}$, i.e., the unit circle. In particular, this proves that, for eigenvalues not in the interior of the $L^2$-spectrum, the zeroes of these functions do not accumulate at the boundary circle. Hence the polyspherical functions can be used to normalise the $n^{\text{th}}$-order Poisson kernels. By this tool, we extend to this setting several classical results of potential theory: namely, we study the boundary behaviour of $λ$-polyharmonic functions, starting with Dirichlet and Riquier type problems and then proceeding to Fatou type admissible boundary limits.

math.FA↗

arXiv2022

Moments of Riesz measures on Poincaré disk and homogeneous tree -- a comparative study

One of the purposes of this paper is to clarify the strong analogy between potential theory on the open unit disk and the homogeneous tree, to which we dedicate an introductory section. We then exemplify this analogy by a study of Riesz measures. Starting from interesting work by Favorov and Golinskii [A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators. Linear and complex analysis, pp. 37-47, Amer. Math. Soc. Transl. (2) 226, Amer. Math. Soc., Providence, RI, 2009], we consider subharmonic functions on the open unit disk, resp. on the homogenous tree. Supposing that we can control the way how those functions may tend to infinity at the boundary, we derive moment type conditions for the Riesz measures. One one hand, we generalise the previous results for the disk, and on the other hand, we show how to obtain analogous results in the discrete setting of the tree.

math.AP↗

arXiv2022

Limit distributions of branching Markov chains

We study branching Markov chains on a countable state space (space of types) $\mathscr{X}$, with the focus on the qualitative aspects of the limit behaviour of the evolving empirical population distributions. No conditions are imposed on the multitype offspring distributions at the points of $\mathscr{X}$ other than to have the same average and to satisfy a uniform $L \log L$ moment condition. We show that the arising population martingale is uniformly integrable. Convergence of population averages of the branching chain is then put in connection with stationary spaces of the associated ordinary Markov chain on $\mathscr{X}$ (assumed to be irreducible and transient). This is applied, in particular, to the boundaries of appropriate compactifications of $\mathscr{X}$. Final considerations consider the general interplay between the measure theoretic boundaries of the branching chain and the associated ordinary chain.

math.PR↗

arXiv2022

Networks with complex weights: Green function and power series

We introduce a Green function and analogues of other related kernels for finite and infinite networks whose edge weights are complex-valued admittances with positive real part. We provide comparison results with the same kernels associated with corresponding reversible Markov chains, i.e., where the edge weights are positive. Under suitable conditions, these lead to comparison of series of matrix powers which express those kernels. We show that the notions of transience and recurrence extend by analytic continuation to the complex-weighted case even when the network is infinite. Thus, a variety of methods known for Markov chains extend to that setting.

math-ph↗

arXiv2021

Ratio limits and Martin boundary

Consider an irreducible Markov chain which satisfies a ratio limit theorem, and let $ρ$ be the spectral radius of the chain. We investigate the relation of the the $ρ\,$-Martin boundary with the boundary induced by the $ρ\,$-harmonic kernel which appears in the ratio limit. Special emphasis is on random walks on non-amenable groups, specifically, free groups and hyperbolic groups.

math.PR↗

arXiv2021

Laplace and bi-Laplace equations for directed networks and Markov chains

The networks of this -- primarily (but not exclusively) expository -- compendium are strongly connected, finite directed graphs $X$, where each oriented edge $(x,y)$ is equipped with a positive weight (conductance) $a(x,y)$. We are not assuming symmetry of this function, and in general we do not require that along with $(x,y)$, also $(y,x)$ is an edge. The weights give rise to a difference operator, the normalised version of which we consider as our Laplace operator. It is associated with a Markov chain with state space $X$. A non-empty subset of $X$ is designated as the boundary. We provide a systematic exposition of the different types of Laplace equations, starting with the Poisson equation, Dirichlet problem and Neumann problem. For the latter, we discuss the definition of outer normal derivatives. We then pass to Laplace equations involving potentials, thereby also addressing the Robin boundary problem. Next, we study the bi-Laplacian and associated equations: the iterated Poisson equation, the bi-Laplace Neumann and Dirichlet problems, and the "plate equation". It turns out that the bi-Laplace Dirichlet to Neumann map is of non-trivial interest. The exposition concludes with two detailed examples.

math.PR↗

arXiv2021

Recurrence of 2-dimensional queueing processes, and random walk exit times from the quadrant

Let $X = (X_1, X_2)$ be a 2-dimensional random variable and $X(n), n \in \mathbb{N}$ a sequence of i.i.d. copies of $X$. The associated random walk is $S(n)= X(1) + \cdots +X(n)$. The corresponding absorbed-reflected walk $W(n), n \in \mathbb{N}$ in the first quadrant is given by $W(0) = x \in \mathbb{R}_+^2$ and $W(n) = \max \{ 0, W(n-1) - X(n) \}$, where the maximum is taken coordinate-wise. This is often called the Lindley process and models the waiting times in a two-server queue. We characterize recurrence of this process, assuming suitable, rather mild moment conditions on $X$. It turns out that this is directly related with the tail asymptotics of the exit time of the random walk $x + S(n)$ from the quadrant, so that the main part of this paper is devoted to an analysis of that exit time in relation with the drift vector, i.e., the expectation of $X$.

math.PR↗

arXiv2019

Boundary behaviour of $λ$-polyharmonic functions on regular trees

This paper studies the boundary behaviour of $λ$-polyharmonic functions for the simple random walk operator on a regular tree, where $λ$ is complex and $|λ|> ρ$, the $\ell^2$-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved and a non-tangential Fatou theorem is proved.

math.PR↗

arXiv2019

The language of self-avoiding walks

Let $X=(V\!X,E\!X)$ be an infinite, locally finite, connected graph without loops or multiple edges. We consider the edges to be oriented, and $E\!X$ is equipped with an involution which inverts the orientation. Each oriented edge is labelled by an element of a finite alphabet $\mathbfΣ$. The labelling is assumed to be deterministic: edges with the same initial (resp. terminal) vertex have distinct labels. Furthermore it is assumed that the group of label-preserving automorphisms of $X$ acts quasi-transitively. For any vertex $o$ of $X$, consider the language of all words over $\mathbfΣ$ which can be read along self-avoiding walks starting at $o$. We characterize under which conditions on the graph structure this language is regular or context-free. This is the case if and only if the graph has more than one end, and the size of all ends is $1$, or at most $2$, respectively.

math.CO↗

arXiv2019

Multiple boundary representations of $λ$-harmonic functions on trees

We consider a countable tree $T$, possibly having vertices with infinite degree, and an arbitrary stochastic nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with eigenvalue $λ\in \mathbb{C}$, under the condition that the oriented edges can be equipped with complex-valued weights satisfying three natural axioms. These axioms guarantee that one can construct a $λ$-Poisson kernel. The boundary integral is with respect to distributions, that is, elements in the dual of the space of locally constant functions. Distributions are interpreted as finitely additive complex measures. In general, they do not extend to $σ$-additive measures: for this extension, a summability condition over disjoint boundary arcs is required. Whenever $λ$ is in the resolvent of $P$ as a self-adjoint operator on a naturally associated $\ell^2$-space and the diagonal elements of the resolvent (`Green function') do not vanish at $λ$, one can use the ordinary edge weights corresponding to the Green function and obtain the ordinary $λ$-Martin kernel. We then consider the case when $P$ is invariant under a transitive group action. In this situation, we study the phenomenon that in addition to the $λ$-Martin kernel, there may be further choices for the edge weights which give rise to another $λ$-Poisson kernel with associated integral representations. In particular, we compare the resulting distributions on the boundary. The material presented here is closely related to the contents of our `companion' paper arXiv:1802.01976

math.FA↗

arXiv2019

Polyharmonic functions for finite graphs and Markov chains

On a finite graph with a chosen partition of the vertex set into interior and boundary vertices, a $λ$-polyharmonic function is a complex function $f$ on the vertex set which satisfies $(λ\cdot I - P)^n f(x) = 0$ at each interior vertex. Here, $P$ may be the normalised adjaceny matrix, but more generally, we consider the transition matrix $P$ of an arbitrary Markov chain to which the (oriented) graph structure is adapted. After describing these `global' polyharmonic functions, we turn to solving the Riquier problem, where $n$ boundary functions are preassigned and a corresponding `tower' of $n$ successive Dirichlet type problems are solved. The resulting unique solution will be polyharmonic only at those points which have distance at least $n$ from the boundary. Finally, we compare these results with those concerning infinite trees with the end boundary, as studied by Cohen, Colonnna, Gowrisankaran and Singman, and more recently, by Picardello and Woess.

math.PR↗

arXiv2019

Oscillating heat kernels on ultrametric spaces

Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $B \mapsto C(B)$ defined on the collection of all non-singleton balls $B$ of $X$, we consider the associated hierarchical Laplacian $L=L_{C}\,$. The operator $L$ acts in $\mathcal{L}^{2}(X,m),$ is essentially self-adjoint and has a pure point spectrum. It admits a continuous heat kernel $\mathfrak{p}(t,x,y)$ with respect to $m$. We consider the case when $X$ has a transitive group of isometries under which the operator $L$ is invariant and study the asymptotic behaviour of the function $t\mapsto \mathfrak{p}(t,x,x)=\mathfrak{p}(t)$. It is completely monotone, but does not vary regularly. When $X=\mathbb{Q}_{p}\,$, the ring of $p$-adic numbers, and $L=\mathcal{D}^α $, the operator of \ fractional derivative of order $α,$ we show that $\mathfrak{p}(t)=t^{-1/α}\mathcal{A}% (\log_{p}t)$, where $\mathcal{A}(τ)$ is a continuous non-constant $α$-periodic function. We also study asymptotic behaviour of $\min\mathcal{A}$ and $\max\mathcal{A}$ as the space parameter $p$ tends to $\infty$. When $X=S_{\infty}\,$, the infinite symmetric group, and $L$ is a hierarchical Laplacian with metric structure analogous to $\mathcal{D}^α,$ we show that, contrary to the previous case, the completely monotone function $\mathfrak{p}(t)$ oscillates between two functions $ψ(t)$ and $Ψ(t)$ such that $ψ(t)/Ψ(t)\to 0$ as $t \to \infty\,$.

math.PR↗

arXiv2018

Boundary representations of $λ$-harmonic and polyharmonic functions on trees

On a countable tree $T$, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with eigenvalue $λ\in \mathbb{C}$. This is possible whenever $λ$ is in the resolvent set of $P$ as a self-adjoint operator on a suitable $\ell^2$-space and the on-diagonal elements of the resolvent ("Green function") do not vanish at $λ$. We show that when $P$ is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition holds for all $λ\ne 0$ in the resolvent set. These results extend and complete previous results by Cartier, by Figà-Talamanca and Steger, and by Woess. For those eigenvalues, we also provide an integral representation of $λ$-polyharmonic functions of any order $n$, that is, functions $f: T \to \mathbb{C}$ for which $(λ\cdot I - P)^n f=0$. This is a far-reaching extension of work of Cohen et al., who provided such a representation for simple random walk on a homogeneous tree and eigenvalue $λ=1$. Finally, we explain the (much simpler) analogous results for "forward only" transition operators, sometimes also called martingales on trees.

math.FA↗