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 $\rho_* = e^{h(\mu)}$, where $h(\mu)$ is the asymptotic entropy of the underlying random walk on $G$ with step distribution $\mu$. Below this value $\rho_*$, 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.
We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution $\mu$ is critical and attracted to an $\alpha$-stable law. This is done via the infinite invariant BRW introduced by Le Gall&Lin and a new criterion which relates this growth rate of the range to a notion of dimension of the underlying tree in a general way. Then, in the transient case (that is, when the range does grow linearly), we extend the notion of branching capacity to this $\alpha$-stable case. We show that it is still related to the asymptotic probability that a BRW (or its infinite version) reaches a distant set in $\mathbb Z^d$, and we estimate the $\alpha$-stable branching capacity of balls.
We study first-passage percolation on the $\ell$-spread-out one-dimensional cycle of size $n$, where vertices are connected if their graph distance is at most $\ell$. We assign i.i.d.~non-negative random weights from a Weibull distribution $\omega_e \sim \mathrm{Exp}(1)^{1/\theta}$ to the edges for $\theta>0$ fixed. This paper investigates the transition in the asymptotic behavior of the passage time $T_n$ between two typical vertices and the hop-count of the optimal path as the connectivity parameter $\ell$ diverges with $n$. We identify two fundamentally distinct geometric regimes. In the mesoscopic regime ($1 \ll \ell \ll n$), the optimal path locally mimics a spatial branching random walk but remains globally constrained to a one-dimensional geometry. We establish a law of large numbers characterized by the front speed of a Crump--Mode--Jagers branching random walk, prove a central limit theorem with Gaussian fluctuations when $\ell\ll n^{1/4}$, and show that the expected hop-count grows proportionally with the spatial distance. In the macroscopic regime ($\ell \approx \lambda n$ for $\lambda \in (0,1/2)$), the graph becomes a highly connected mean-field network. We prove that the passage time collapses to a $\log n$ scale with constant order non-Gaussian fluctuations, explicitly determining the extreme-value limit driven by the collision of two independent non-spatial CMJ processes. We establish a law of large numbers for the hop-count. Finally, we rigorously trace the transition in the order of the mean of $T_n$ between these two regimes, demonstrating an order transition for the passage time across the critical connectivity threshold $\ell \asymp n/\log n$. Our results provide a comprehensive deterministic-range interpolation from spatial Gaussian fluctuations to mean-field extreme-value fluctuations.
This paper investigates the global geometry of the group of Hamiltonian diffeomorphisms $\operatorname{Ham}(M,\omega)$ using random walks. On a large class of symplectic manifolds, we show that the expected Hofer norm of such a random walk grows at least as fast as the square root of the number of steps. Furthermore, we show that if the random walk is restricted to an abelian subgroup, then the growth rate is also bounded from above by the square root of the number of steps. We also provide some numerical evidence suggesting that this upper bound fails away from commutative subgroups. This provides, for subgroups of the group of Hamiltonian diffeomorphisms, a probabilistic version of the flatness observed in commutative finite-dimensional Lie groups. En route, we show that the class of probability measures introduced in the prequel is a class of Borel measures with respect to the $C^\infty$-topology on $\operatorname{Ham}(M,\omega)$, show the measurability of the stable commutator length, and show a central limit theorem for Hofer-Lipschitz quasimorphisms on $\operatorname{Ham}(M,\omega)$.
We study self-avoiding walks on the canonical one-sided graphs of Lindstrom nested fractals. We prove that the connective constant $\mu$ exists and identify $\log\mu$ with the critical inverse temperature of a finite-dimensional boundary-state renormalization. If the boundary-state partition vectors are bounded at criticality, then the fixed-length counts $c_n$ satisfy two-sided polynomial bounds around $\mu^n$. We also prove that $h$-flexibility implies $c_{n+h}/c_n\to\mu^h$. For regular polygonal $N$-gaskets, we derive exact crossing recursions, determine the smallest flexibility step $h$, and obtain explicit algebraic connective constants for the $6$- and $9$-gaskets. The Vicsek graph has no flexibility step, and its successive ratios do not converge.
We determine the critical probability for Bernoulli bond percolation on essentially any finite graph. Namely, letting $\lambda(G)$ denote the spectral radius (maximum eigenvalue) of $G$, we prove that the critical probability is at $1/\lambda(G)$: above this probability there is typically a component of order $\Omega(\lambda(G))$, whereas below it all components are of order at most $O(\sqrt{|G|})$. These results in particular confirm a conjecture of Krivelevich and Samotij about percolation on graphs of a given average degree, and vastly extend theorems of Bollob\'as, Borgs, Chayes, and Riordan, who proved analogous results but only for dense graphs. Our theorems are optimal in many regimes, and also demonstrate that percolation has an unexpectedly subtle behaviour on graphs whose spectral radius is roughly the square root of their maximum degree.
We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.
Philipp Hake, Matthias Keller, Felix Pogorzelski· 0 citations
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