H. Berestycki and C. Graham (2022) proved that the F-KPP reaction-diffusion equation $\partial_t u = \frac{1}{2} \Delta u + u(1-u)$ in the half-place with Dirichlet boundary conditions admit traveling waves for all speed $c \geq \sqrt{2}$. Using the duality between this PDE and the branching Brownian motion in the half-plane with absorption at the boundary, we prove that the minimal speed traveling wave is in fact unique (up to shift). Moreover, we give a probabilistic representation of this traveling wave $\Phi$ in terms of the Laplace transform of a certain "derivative martingale" of this branching Brownian motion. 

We use this probabilistic representation to describe the asymptotic behavior of $\Phi$ away from the boundary of the domain, proving that 

\[ \lim_{y \to \infty} \Phi\left(x + \frac{1}{\sqrt{2}} \log y, y\right) = w(x) \] 

where $w$ is the usual one-dimensional traveling wave. This talk is based on joint work with Julien Berestycki, Cole Graham and Yujin H. Kim.

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In two and three dimensions, the scaling limit of the uniform spanning tree of Z^d is a measured, rooted spatial tree, and its embedding in R^d fills the space. For these integer lattices and high-dimensional finite graphs, the proofs of the existence of the corresponding limit trees rely on Wilson's algorithm. The situation is different in Z^d for d ≥ 5. In this case, the infinite-volume limit of uniform spanning trees of finite subgraphs is the uniform spanning forest (USF) with, almost surely, infinitely many trees. Sampling with Wilson's algorithm, one builds different trees simultaneously. Hence, studying the scaling limit of one tree in the USF requires a different approach. This talk will present recent advances in this direction for the USF of Z^d, d ≥ 5, based on ongoing joint work with Tom Hutchcroft. 

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In this talk, we  present some recent results on the recurrence properties and the asymptotic behavior of the « oscillating random walk on Z ». We propose a simple criteria for recurrence and state an invariance principle for this process. The main tool is a renewal theorem for aperiodic sequences of operators, due to S. Gouezel.

We call a decoupled random walk a sequence $\hat S_1$, $\hat S_2,\ldots$ of independent random variables such that, for each $n\in\mathbb{N}$, $\hat S_n$ has the same distribution as the position at time $n$ of a standard random walk with nonnegative jumps. Similarly, we call a decoupled renewal process the counting process $(\hat N(t))_{t\geq 0}$ defined by $\hat N(t)=\sum_{n\geq 1}\1_{\{\hat S_n\leq t\}}$. I shall present a functional limit theorem for $(\hat N(t))_{t\geq 0}$, properly scaled, normalized and centered, as $t\to\infty$ under the assumption that the variance of $\hat S_1$ is positive and finite. Also, I shall discuss the asymptotic of $\log \mathbb{P}\{\min_{n\geq 1}\hat S_n>t\}$ as $t\to\infty$ under various assumptions imposed on the distribution of $\hat S_1$. Our interest to the so defined decoupled random walks was caused by their appearance in the particular case when $\hat S_1$ has an exponential distribution of unit mean in the context of infinite Ginibre point processes. The talk is based on a recent joint work with Gerold Alsmeyer and Zakhar Kabluchko (Muenster).

We consider autoregressive sequences $X_n=aX_{n-1}+\xi_n$ and $M_n=\max\{aM_{n-1},\xi_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{\xi_k\}$. It is known that if $\mathbb P(\xi_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index $-1-d/\log a$. We also prove limit theorems for $\{X_n\}$ and $\{M_n\}$ conditioned to stay over a fixed level $x_0$. Furthermore, we study tail asymptotics for recurrence times of $\{X_n\}$ and $\{M_n\}$ in the case when these chains are positive recurrent and the tail of $\log\xi_1$ is subexponential.

We consider a random walk on a regular tree, where each edge is assigned a random weight or conductance. The random walk crosses an edge with probability proportional to the edge conductance and additionally, conductances of edges leading away from the root are increased by a bias factor. The distance to the root then satisfies a law of large numbers with limit the effective speed of the random walk, which depends in a complicated way on the conductance law and bias parameter. We are interested in properties of the speed as a function of the bias. For example, is the speed monotone increasing in the strength of the bias? This talk is based on joint work with Nina Gantert and Yuki Tokushige. 

In the talk we will present laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our first result is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we will briefly show how to compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero-one law for functionals of convex hulls of walks with drift, of some independent interest.

This is a joint project with Nikola Sandrić, Stjepan Šebek and Andrew Wade