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Martin Klötzer

Publications and source records attributed to Martin Klötzer.

5 recordsLinked to original sources
arXiv2026

Phase transition for the asymptotic entropy of branching random walks on groups

We consider supercritical branching random walks (BRW) on countable groups $G$ and we prove that the asymptotic entropy of the empirical distributions of the BRW has a phase transition at $ρ_* = e^{h(μ)}$, where $h(μ)$ is the asymptotic entropy of the underlying random walk on $G$ with step distribution $μ$. Below this value $ρ_*$, the asymptotic empirical entropy of BRW equals the logarithm of the exponential growth rate of the population. Above this value, it is constantly equal to the asymptotic entropy of the underlying random walk. In particular, this answers questions from Kaimanovich-Woess [MR4663513, Section 6.3] about the existence and the behavior of the asymptotic entropy.

math.PR↗
arXiv2026

Maximal and minimal displacement of supercritical branching random walks on free products of groups

We prove that the maximal and minimal displacement of branching random walks with mean offspring number $ρ>1$ on free products of finite groups grows linearly almost surely. More precisely, we establish that the linear speed for the maximal (respectively minimal) displacement is given by the largest (respectively smallest) intersection point of the large deviation rate function of the underlying random walk with the horizontal line at height $\logρ$. The proof is based on constructing an associated multitype branching process which consists of particles that travel fast enough, and distinguishing the types via the suffix of the particles locations.

math.PR↗
arXiv2026

Relative stationary dynamical systems

Let $G$ be a locally compact second countable group equipped with an admissible non-degenerate Borel probability measure $μ$. We generalize the notion of $μ$-stationary systems to $μ$-stationary $G$-factor maps $π: (X,ν)\to (Y,η)$. For these stationary relations between dynamical systems, we provide a structure theorem, which generalizes the structure theorem of Furstenberg-Glasner. Furthermore, we show the existence and uniqueness of a relative version of the Poisson boundary in this setup.

math.DS↗
arXiv2026

Stabilization of stochastic networks in Markovian environment

We establish criteria under which stochastic networks in a Markovian environment stabilize, thus confirming Conjecture 7.2 from Levine-Greco [GL23]. The networks evolve on finite connected graphs $G=(V,E)$, and their dynamics are encoded by $V \times V$ toppling matrices $M$, whose columns record the expected number of topplings when the environment is in stationarity. Stabilization and non-stabilization are characterized by a parameter $ρ$ which depends on the largest eigenvalue of the matrix $M+αI$, with $α=1+\max\{-M(v,v):v\in V\}$. The proofs rely on the toppling random walk, in which toppled vertices are sampled according to the eigenvector associated with the largest eigenvalue of $M$.

math.PR↗