We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps \eta_i's of the RW,...

While records and order statistics of independent and identically distributed (i.i.d.) random variables X_1, ..., X_N are fully understood, much less is known for strongly correlated random variables, which is often the situation encountered in statistical physics. Recently, it was shown, in a...

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length \eta drawn from a continuous distribution f(\eta) which is symmetric around a constant drift...

We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and...

We reconsider the problem of optimal time to sell a stock studied recently by Shiryaev, Xu and Zhou using path integral methods. This method allows us to confirm the results obtained by these authors and extend them to a parameter region inaccessible to the method used by Shiryaev et. al. We...

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x0=0,x1,x2,…,xn} up to n steps where xi represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a...

We introduce and solve exactly a family of invariant 2×2 random matrices, depending on one parameter η, and we show that rotational invariance and real Dyson index β are not incompatible properties. The probability density for the entries contains a weight function and a multiple...

An intriguing connection between extreme value statistics and traveling fronts has been found recently in a number of diverse problems. In this short review we outline a few such problems and consider their various applications.