We consider random walks amongst random conductances in the cases where the conductances can be arbitrarily small, with a heavy-tailed distribution at 0, and where the conductances may or may not have a heavy-tailed distribution at infinity. We study the long time behaviour of these processes and prove aging statements. When the heavy tail is only at 0, we prove that aging can be observed for the maximum of the process, i.e. the same maximal value is attained repeatedly over long time-scales. When there are also heavy tails at infinity, we prove a classical aging result for the position of the walker, as well as a sub-aging result that occurs on a shorter time-scale.

We consider Bienaymé-Galton-Watson trees in random environment, where each generation k is attributed a random offspring distribution μk, and (μk)k≥0 is a sequence of independent and identically distributed random probability measures. We work in the "strictly critical" regime where, for all k, the average of μk is assumed to be equal to 1 almost surely, and the variance of μk has finite expectation. We prove that, for almost all realizations of the environment (more precisely, under some deterministic conditions that the random environment satisfies almost surely), the scaling limit of the tree in that environment, conditioned to be large, is the Brownian continuum random tree. The habitual techniques used for standard Bienaymé-Galton-Watson trees, or trees with exchangeable vertices, do not apply to this case. Our proof therefore provides alternative tools.

We study a non-reversible random walk advected by the symmetric simple exclusion process, so that the walk has a local drift of opposite sign when sitting atop an occupied or an empty site. We prove that the back-tracking probability of the walk exhibits a sharp transition as the density ρ of particles in the underlying exclusion process varies across a critical density ρ_c. Our results imply that the speed v=v(ρ) of the walk is a strictly monotone function and that the zero-speed regime is either absent or collapses to a single point, ρ_c, thus solving a conjecture of arXiv:1906.03167. The proof proceeds by exhibiting a quantitative monotonicity result for the speed of a truncated model, in which the environment is renewed after a finite time horizon L. The truncation parameter L is subsequently pitted against the density ρ to carry estimates over to the full model. Our strategy is somewhat reminiscent of certain techniques recently used to prove sharpness results in percolation problems. A key instrument is a combination of renormalisation arguments with refined couplings of environments at slightly different densities, which we develop in this article. Our results hold in fact in greater generality and apply to a class of environments with possibly egregious features, outside perturbative regimes.

In this article, we prove a lower bound for the fluctuations of symmetric random walks on dynamic random environments in dimension $1 + 1$ in the perturbative regime where the walker is weakly influenced by the environment. We suppose that the random environment is invariant with respect to translations and reflections, satisfies the FKG inequality and a mild mixing condition. The techniques employed are inspired by percolation theory, including a Russo-Seymour-Welsh (RSW) inequality. To exemplify the generality of our results, we provide two families of fields that satisfy our hypotheses: a class of Gaussian fields and Confetti percolation models.

In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes $N$ and $F$ in a finite graph $\mathcal G$. The ants perform a sequence of random walks on this graph, starting from the nest and stopped when first hitting the source of food. At each step of its random walk, the $n$-th ant chooses to cross a neighbouring edge with probability proportional to the number of preceding ants that crossed that edge at least once. We say that {\it the ants find the shortest path} if, almost surely as the number of ants grow to infinity, almost all the ants go from the nest to the source of food through one of the shortest paths, without loosing time on other edges of the graph. Our contribution is three-fold: (1) We prove that, if $\mathcal G$ is a tree rooted at $N$ whose leaves have been merged into node $F$, and with one edge between $N$ and $F$, then the ants indeed find the shortest path. (2) In contrast, we provide three examples of graphs on which the ants do not find the shortest path, suggesting that in this model and in most graphs, ants do not find the shortest path. (3) In all these cases, we show that the sequence of normalised edge-weights converge to a {\it deterministic} limit, despite a linear-reinforcement mechanism, and we conjecture that this is a general fact which is valid on all finite graphs. To prove these results, we use stochastic approximation methods, and in particular the ODE method. One difficulty comes from the fact that this method relies on understanding the behaviour at large times of the solution of a non-linear, multi-dimensional ODE.

We study the so-called frog model on Z with two types of lazy frogs, with parameters p(1), p(2) is an element of (0, 1] respectively, and a finite expected number of dormant frogs per site. We show that for any such p(1) and p(2) there is positive probability that the two types coexist (i.e. that both types activate infinitely many frogs). This answers a question of Deijfen, Hirscher, and Lopes in dimension one.

Abstract We study the so-called frog model on ${\mathbb{Z}}$ with two types of lazy frogs, with parameters $p_1,p_2\in (0,1]$ respectively, and a finite expected number of dormant frogs per site. We show that for any such $p_1$ and $p_2$ there is positive probability that the two types coexist (i.e. that both types activate infinitely many frogs). This answers a question of Deijfen, Hirscher, and Lopes in dimension one.

It is well known in biology that ants are able to find shortest paths between their nest and the food by successive random explorations, without any mean of communication other than the pheromones they leave behind them. This striking phenomenon has been observed experimentally and modelled by different mean-field reinforcement-learning models in the biology literature. In this paper, we introduce the first probabilistic reinforcement-learning model for this phenomenon. In this model, the ants explore a finite graph in which two nodes are distinguished as the nest and the source of food. The ants perform successive random walks on this graph, starting from the nest and stopping when they first reach the food; the transition probabilities of each random walk depend on the realizations of all previous walks through some dynamic weighting of the graph. We discuss different variants of this model based on different reinforcement rules and show that slight changes in this reinforcement rule can lead to drastically different outcomes. We prove that the ants indeed eventually find the shortest path(s) between their nest and the food in two variants of this model and when the underlying graph is, respectively, any series-parallel graph and a five-edge nonseries-parallel losange graph. Both proofs rely on the electrical network method for random walks on weighted graphs and on Rubin's embedding in continuous time. The proof in the series-parallel cases uses the recursive nature of this family of graphs, while the proof in the seemingly simpler losange case turns out to be quite intricate: it relies on a fine analysis of some stochastic approximation, and on various couplings with standard and generalised Polya urns.

We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $\mathbb{Z}\times \Gamma$, with $\Gamma$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $\rho \in [0, 1]$ of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $\rho$ except for at most two values $\rho_-, \rho_+ \in [0, 1]$. The asymptotic speed we obtain in our LLN is a monotone function of $\rho$. Also, $\rho_-$ and $\rho_+$ are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is $1/2$ and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. Finally, we prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

We present a simple model of a random walk with partial memory, which we call the random memory walk. We introduce this model motivated by the belief that it mimics the behavior of the once-reinforced random walk in high dimensions and with small reinforcement. We establish the transience of the random memory walk in dimensions three and higher, and show that its scaling limit is a Brownian motion.

The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define a quantity, that we call the branching-ruin number of a tree, which provides (in the spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measure trees with polynomial growth. We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk. We define a sharp and effective (i.e. computable) criterion characterizing the recurrence/transience of a larger class of self-interacting walks on trees, providing the complete picture for their phase transition.

We study once-reinforced biased random walk on \(\mathbb {Z}^d\). We prove that for sufficiently large bias, the speed \(v(\beta )\) is monotone decreasing in the reinforcement parameter \(\beta \) in the region \([0,\beta _0]\), where \(\beta _0\) is a small parameter depending on the underlying bias. This result is analogous to results on Galton–Watson trees obtained by Collevecchio and the authors.

The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees. In this paper, we continue the investigation of the link between the branching-ruin number and the criticality of random processes on trees. First, we study random walks on random conductances on trees, when the conductances have an heavy tail at $0$, parametrized by some $p>1$, where $1/p$ is the exponent of the tail. We prove a phase transition recurrence/transience with respect to $p$ and identify the critical parameter to be equal to the branching-ruin number of the tree. Second, we study a multi-excited random walk on trees where each vertex has $M$ cookies and each cookie has an infinite strength towards the root. Here again, we prove a phase transition recurrence/transience and identify the critical number of cookies to be equal to the branching-ruin number of the tree, minus 1. This result extends a conjecture of Volkov (2003). Besides, we study a generalized version of this process and generalize results of Basdevant and Singh (2009).

In this note we show a simple formula for the joint density of local times, last exit tree and cycling numbers of continuous-time Markov chains on finite graphs, which involves the modified Bessel function of the first type.

We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative oncereinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.

We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $\mathbb{Z}\times Γ$, with $Γ$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.

We consider biased random walks in positive random conductances on the d-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional Law of Large Numbers for the position of the walker, properly rescaled. Moreover, we state a functional Central Limit Theorem where an atypical process, related to the Fractional Kinetics, appears in the limit.

In this short paper, we consider the Once-reinforced random walk with reinforcement parameter a on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter a(0) such that the Once-reinforced random walk is almost surely recurrent if a > a(0) and almost surely transient if a < a(0). This provides the first examples of phase transition for the Once-reinforced random walk.

In this short paper, we consider the Once-reinforced random walk with reinforcement parameter a on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter a0 such that the Once-reinforced random walk is almost surely recurrent if a > a0 and almost surely transient if a < a0. This provides the first examples of phase transition for the Once-reinforced random walk.