In this paper, we study the asymptotic behaviours of a critical branching random walk in $\mathbb{R}^d$ under the assumption that the offspring distribution belongs to the domain of attraction of an $\alpha$-stable law with $\alpha\in(1,2]$, and that the jump distribution has a finite $\frac{2\alpha}{\alpha-1}$-th moment. First, we establish the precise decay rate for the tail probability of the all-time maximal displacement $M^d$. Next, we investigate the maximal displacement $M_n^d$ at generation $n$ and prove a conditional limit theorem for the distribution of $M_n^d$ given that the process survives up to generation $n$. These results extend the corresponding 1-dimensional results of Lalley and Shao (2015) to the case $d\ge2$. Finally, we study the asymptotic behaviour of the total progeny $\zeta$. In particular, we show that, conditioned on the event $\{M^d\ge x\}$, $\zeta$ converges in distribution under an appropriate normalization. This result reveals a quantitative relationship between the maximal displacement and the total progeny size.
In this paper, we study a bootstrap percolation process on the finite triangular grid $\mathfrak{T}_n$ of side length $n$. We say that a subset $\eta$ of points in $\mathfrak{T}_n$ percolates if the final configuration, starting from $\eta$, is the whole grid $\mathfrak{T}_n$. A basis of size $n$ is then a subset of points of $\mathfrak{T}_n$ of minimum cardinality which percolates. In this paper, we first prove that the generating function counting bases satisfies an algebraic differential equation. Then, by analysing a modified version of this equation, we prove that the number $t_n$ of bases of size $n$ exhibits a stretched exponential asymptotic behaviour. More precisely, we show that $t_n \sim c n!e^{\sqrt{12n}}n^{5/12}$, for some constant $c>0$. These bases were recently shown by the second author to be in bijection with $3$-permutations avoiding the patterns $(12, 12)$ and $(231, 312)$, so this represents to our knowledge the first proven example of an asymptotic stretched exponential appearing in the study of pattern avoiding permutations.
A. Price, Juliette Schabanel, Paul Th'evenin· 1 citation
We consider a stochastic process on $\mathbb{Z}^d$ for $d \geq 1$. Given a translation invariant and ergodic starting configuration of finite clusters, each cluster $C$ performs a continuous time simple random walk with rate $|C|^{-\alpha}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. In all dimensions, we show that if $\alpha\ge 0$, there is almost surely no spontaneous creation of an infinite cluster within finite time. Moreover, for any $\alpha\le-1-2/d$ there is a finite-time blowup almost surely. In the regime $\alpha\in(-1,0)$ we show that the behavior greatly depends on the initial configuration. In addition, in dimension one, we establish the exact phase diagram.
Noam Berger, Eviatar B. Procaccia, Dominik Schmid et al.· 0 citations
We consider the weight-dependent random connection model on a Poisson point process of intensity $\lambda$ on $\mathbb{R}\times(0,1)$ in which the vertices $(x,t)$ and $(y,s)$ are joined precisely when $(t\vee s)|x-y|\le\beta$. Points at distance $d$ are joined with probability $\min(1,\beta/d)^2$, the critical decay of one-dimensional long-range percolation, and edges sharing a vertex are dependent through the common mark. We prove that the model has a genuine phase transition: for $\lambda\beta<1$ almost surely all connected components are finite, while for $\lambda\beta\ge31$ an infinite component exists, so at intensity one the critical value satisfies $\beta_c\in[1,31]$; a numerical study included as an appendix places it near $2$. By kernel and profile comparisons the supercritical bound extends to the age-dependent random connection model on the line, which with indicator profile has a non-degenerate phase transition at every value of its parameter, closing a case of the one-dimensional phase diagram left open in earlier work. The lower bound is proved by disconnecting nested pairs of long edges ("rainbows") with cut-point certificates, an argument developed first in a discrete skeleton of the model with the vertices pinned to $\mathbb{Z}$. The skeleton is of independent interest: it has no supercritical phase at all, jumping from total fragmentation to trivial connectivity even though almost surely infinitely many edges cross every fixed site. The supercritical argument is a Peierls argument on the binary tiling of the hyperbolic half-plane.
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.
Jérémie Brieussel, R. Kaiser, Martin Klötzer et al.· 0 citations
We study the frog model on $\mathbb Z^d$ and on the discrete tori $\mathbb T_L^d$, $d\ge 2$, with a symmetric, translation-invariant, and heavy-tailed transition kernel satisfying \[ Q(x,y)\asymp |x-y|^{-(d+\alpha)}, \qquad \alpha>0. \] Starting from an i.i.d. Poisson$(\lambda)$ number of sleeping particles per site and one active particle at the origin. Active particles perform independent $Q$-random walks and activate the particles they encounter. We first determine the timescale for activating distant vertices. When $\alpha\in(0,d)$, the time required to activate all vertices within distance $L$ of the origin is, with high probability, \[ (\log L)^{\Delta+o(1)}, \qquad \Delta^{-1}:=\log_2\left(\frac{2d}{d+\alpha}\right), \] as $L\to\infty$. This polylogarithmic spreading contrasts sharply with the linear spreading of the classical frog model driven by simple random walks; see Alves, Machado, and Popov (2002) and Ram\'irez and Sidoravicius (2004). When $\alpha>d$, we recover this classical linear behavior by proving matching linear upper and lower bounds; at $\alpha=d$, we prove a linear upper bound. Finally, we consider the finite-lifespan model on $\mathbb T_L^d$, in which each particle is removed after taking $\ell$ steps. We show that the cover lifespan, defined as the smallest $\ell$ for which the torus is entirely activated, is asymptotic to the cover time of a Poisson$(\lambda L^d)$ cloud of independent stationary random walkers.
For a Markov kernel $T$ with an invariant probability measure $\pi$, we give a self-contained proof of the Markov chain convergence theorem via a criterion called asymptotic equivalence with the target. It assumes two parts about the Lebesgue decompositions of $T^n_x$ and $\pi$ for every starting point $x$: 1.) asymptotic absolute continuity: the singular mass $\mathrm{sing}(T^n_x \mid \pi)$ tends to $0$; and, 2.) asymptotic domination of the target: the singular mass $\mathrm{sing}(\pi \mid T^n_x)$ tends to $0$, as $n \to \infty$. Assuming a jointly measurable density for the absolutely continuous part of each iterate $T^n$ w.r.t. $\pi$, this criterion is sufficient and necessary for convergence. A positive minorant density version of it is verified in three cases: i.) $T$ has a positive transition density w.r.t. $\pi$; ii.) $T$ consists of an absolutely continuous part with positive transition density together with an atom at the starting point, which covers the Metropolis-Hastings algorithm; iii.) the transition density is positive only after a finite number of steps that may depend on the starting point $x$. To demonstrate our general criterion, we investigate the Gibbs sampler with random scan and the parallel tempering algorithm. Furthermore, we show that in all mentioned settings Birkhoff's ergodic theorem applies, so as to obtain the strong law of large numbers. Throughout this paper, neither irreducibility, nor aperiodicity, nor recurrence, nor couplings, nor splitting constructions, nor small sets are used. In all results, the state space is a general measurable space with no structure beyond a $\sigma$-algebra. That joint measurability is assumed of the Markov kernel, not of the space; countable generation supplies it. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion.