In this talk, I will first review some results on survival probability for random walks and its application to record and extreme value statistics [1]. This survival probability is universal for continuous symmetric random walk and known as Sparre-Andersen formula [2]. As a direct consequence, the record statistics and a large part of the extreme value statistics are universal as well for such random walks. I will then introduce a model of random walk where the sign of consecutive steps are correlated [3]. In particular, in some range of parameters this random walk maps onto a model of active particles, the run-and-tumble particle (RTP). For that walk as well, the survival probability is universal. As an application we compute the record statistics for this model of random walk, which displays the same universality [3]. In particular, we obtain original results in the scaling regime corresponding to the RTP that are quite different from the case of the usual random walk. We also obtain some results for the extreme value statistics of that walk.This work is done in collaboration with F. Mori.
[1] C. Godrèche, S. N. Majumdar, G. Schehr, Record statistics of a strongly correlated time series: random walks and Lévy flights, J. Phys. A 50(33), 333001 (2017).
[2] E. Sparre Andersen, On the fluctuations of sums of random variables II, Math. Scand., 195-223 (1955).
[3] B. Lacroix-A-Chez-Toine, F. Mori, Universal survival probability for a correlatedrandom walk and applications to records, J. Phys. A 9, 101 (2020).