Record statistics for random walk bridges

Claude Godreche, Satya N. Majumdar and Gregory Schehr

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Abstract: We investigate the statistics of records in a random sequence $\{x_B(0)=0,x_B(1),\cdots, x_B(n)=x_B(0)=0\}$ of $n$ time steps. The sequence $x_B(k)$'s represents the position at step $k$ of a random walk `bridge' of $n$ steps that starts and ends at the origin. At each step, the increment of the position is a random jump drawn from a specified symmetric distribution. We study the statistics of records and record ages for such a bridge sequence, for different jump distributions. In absence of the bridge condition, i.e., for a free random walk sequence, the statistics of the number and ages of records exhibits a `strong' universality for all $n$, i.e., they are completely independent of the jump distribution as long as the distribution is continuous. We show that the presence of the bridge constraint destroys this strong `all $n$' universality. Nevertheless a `weaker' universality still remains for large $n$, where we show that the record statistics depends on the jump distributions only through a single parameter $0

Date: 2015-05, Revised 2016-01
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Published in J. Stat. Mech. P07026 (2015)

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