A balanced excited random walk

Abstract

Presented by Marc YorThe following random process on Z 4 is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk step. We prove that this process is almost surely transient. The lower dimensional versions are discussed and various generalizations and related questions are proposed. © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é Nous étudions le processus suivant sur Z 4 . À la première visite en un site, les deux premières coordonnées effectuent un saut d'une marche simple (2-dimensionnelle). Aux visites suivantes en ce site, ce sont les deux dernières coordonnées qui effectuent un saut de marche simple. Nous montrons que ce processus est presque sûrement transitoire. Nous discutons également des dimensions inférieures et divers généralisations et questions connexes sont proposées.

Balanced Excited Random Walk in Two Dimensions

“…Figure 1 shows a simulation of this process, which we also refer to as BERW, up to time n = 10 8 . This model was introduced in more general dimensions d by Benjamini, Kozma and Schapira in [1]: on the first departure from any site a simple random walk step in the first d1 coordinate directions is taken, while on subsequent departures a simple random walk step in the last d2 coordinate directions is taken, and d = d1 + d2. If d1 ∨ d2 ≥ 3 then the walk is trivially transient.…”

“…In the 2 dimensional case 2 = 1 + 1, Benjamini et al [1] conjecture that the walk is recurrent, in the sense that every vertex is visited infinitely often (a.s.). To the best of our knowledge, nothing non-trivial has been proved about this model.…”

“…Therefore, if the growth of Rn is sublinear then the number of vertical steps taken up to time n is sublinear (so the vertical range is o( √ n)) and the limiting shape of the range will be a horizontal line. Benjamini et al [1,Page 2] conjecture that this is the case. Our second main result provides non-trivial lower and upper bounds on the growth of the range and in particular verifies this conjecture.…”

“…We call a sequence x = (xn) n≥0 on Z, a nearest neighbour path if |xn+1 − xn| = 1 for all n ≥ 0. Similarly, we call a sequence z = (z [1] n , z [2] n ) n≥0 in Z 2 a nearest neighbor path if (z [1] n+1 , z [2] n+1 ) − (z [1] n , z [2] n ) 1 = 1 for all n ≥ 0, where • 1 denotes the l 1 norm. A timing sequence is a monotone increasing (north/east) nearest neighbour sequence in Z 2 starting at (0, 0).…”

On recurrence and transience of self-interacting random walks

“…On the first visit to a site the jump of the process has law µ 1 and at further visits it has law µ 2 . The following question was posed in [2]: Is the resulting walk transient?…”