Research

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: 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: We consider a two-elephant walking model in which the elephants interact dynamically. At each time step, each elephant determines its next move randomly based on its partner's past movements. We show that the asymptotic behavior of the elephants mainly depends on the sign and the absolute value of the product of their reinforcement parameters. In various regimes, we establish the law of large numbers and the central limit theorem. Our proofs are based on a connection to the random recursive trees and employ stochastic approximation techniques and martingale methods.

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)}.