Longest interval between zeros of the tied-down random walk, the Brownian bridge and related renewal processes

Abstract

The probability distribution of the longest interval between two zeros of a simple random walk starting and ending at the origin, and of its continuum limit, the Brownian bridge, was analysed in the past by Rosén and Wendel, then extended by the latter to stable processes. We recover and extend these results using simple concepts of renewal theory, which allows to revisit past or recent works of the physics literature.

Fluctuation-dominated phase ordering at a mixed order transition

“…It is interesting to compare the results obtained for the TIDSI model with those obtained for a coarse-grained (CD) depth model. The CD model corresponds to the extreme adiabatic limit for hard-core particles sliding passively on a fluctuating surface [1,2], and can be interpreted as a tied down renewal process on a Brownian bridge, for which analytic calculations can be performed [38,39]. For this model, the distribution of l max can be calculated exactly; as for the TIDSI model, it is a function of l max /L, with multiple mild singularities [38].…”

Two-time correlation and occupation time for the Brownian bridge and tied-down renewal processes

“…The distribution f M (m) is universal, i.e., does not depend on the details of f . We refer to [2] for a study of the distribution of the number of domains N L . The results above also provide some answers to issues raised in the past in the field of stochastically evolving surfaces.…”

“…Tied-down renewal processes with power-law distributions of intervals are generalisations of the Brownian bridge, where an event (or a zero crossing) occurs both at the origin of time and at the final observation time t [1,2]. The Brownian bridge is itself the continuum limit of the tied-down simple random walk, starting and ending at the origin.…”

“…The Brownian bridge is itself the continuum limit of the tied-down simple random walk, starting and ending at the origin. The present work is a sequel of our previous study [2] mainly devoted to the statistics of the longest interval of tied-down renewal processes. Here we investigate further quantities of interest such as the two-time correlation function, the backward and forward recurrence times and the occupation time of the process.…”

“…We shall rely on [2] for some background knowledge, in order to keep the present paper short and avoid repeating the material contained in this reference. Nevertheless, we shall start, in section 2, by giving a brief reminder of the most important definitions needed in the subsequent sections 3-6, devoted respectively to the study of the statistics of the forward and backward recurrence times, the number of renewals between two times, the two-time correlation function and finally the occupation time of the process.…”