Research

Abstract: We prove a conjecture by Bertoin that the multidimensional elephant random walk on Z^d is transient in dimensions d ≥ 3. We show that it undergoes a phase transition in dimensions d = 1, 2, between recurrence and transience at p = (2d+1)/(4d). 

Abstract: Under suitable moment assumptions, we show that a genuinely d-dimensional step-reinforced random walk undergoes a phase transition between recurrence and transience in dimensions d=1,2, and that it is transient for all reinforcement parameters in dimensions d ≥ 3, which solves a conjecture of Bertoin.

Abstract: For the interacting urn model with polynomial reinforcement, it has been conjectured that almost surely one color monopolizes all the urns if the interaction parameter p > 0. We disprove the conjecture. For the case p = 1, we give a sufficient condition for monopoly, which improves a result obtained by Launay. 

Abstract: We introduce the continuous-time vertex-reinforced random walk (cVRRW) as a continuous-time version of the vertex-reinforced random walk (VRRW). On a complete-like graph Kd ∪ ∂Kd, we show that the cVRRW a.s. spends finite time on leaves and equal time on each of the non-leaf vertices up to a convergent error term. For the VRRW, we provide estimates on the rate of convergence of the empirical occupation measure, which, in particular, verifies a conjecture by Limic and Volkov that the true convergence rate is of order n^{-1/(d−1)}.