Narrow Results

We consider the Schrödinger operator with random decaying potential on ${\ell }^2(Z^d)$ and showed that, (i) Integrated Density of States (IDS) coincide with that of free Laplacian in general cases, (ii) the set of extremal eigenvalues, after rescaling, converges to an inhomogeneous Poisson process, under certain condition on the single-site distribution, and (iii) there are “border-line” cases, such that we have Poisson statistics in the sense of (ii) above if the potential does not decay, while we do not if the potential does decay.

Fluctuations of global additive quantities, like total energy or magnetization for instance, can in principle be described by statistics of sums of (possibly correlated) random variables. Yet, it turns out that extreme values (the largest value among a set of random variables) may also play a role in the statistics of global quantities, in a direct or indirect way. This review discusses different connections that may appear between problems of sums and of extreme values of random variables, and emphasizes physical situations in which such connections are relevant. Along this line of thought, standard convergence theorems for sums and extreme values of independent and identically distributed random variables are recalled, and some rigorous results as well as more heuristic reasonings are presented for correlated or non-identically distributed random variables. More specifically, the role of extreme values within sums of broadly distributed variables is addressed, and a general mapping between extreme values and sums is presented, allowing us to identify a class of correlated random variables whose sum follows (generalized) extreme value distributions. Possible applications of this specific class of random variables are illustrated on the example of two simple physical models. A few extensions to other related classes of random variables sharing similar qualitative properties are also briefly discussed, in connection with the so-called BHP distribution.

The cumulative distribution function (CDF) of the maximum in a time series generated by different kinds of deterministic dynamics is computed, in the framework of prototypical models of such dynamics. This CDF is shown to be significantly different in form from that of the maximum in a sequence of identically distributed random variables. In the latter case, the CDF is generically in the domain of attraction of one of the three extreme value distributions. This property is not shared by the CDF for deterministic dynamics. The characteristic features of the CDF are elucidated in the case of periodic, quasiperiodic, fully chaotic and intermittently chaotic dynamics.

We consider the distribution of the maximum for finite, deterministic, periodic and quasiperiodic sequences, and contrast the extreme value distributions in these cases with the classical results for iidrv's. A significant feature in the case of deterministic sequences is a multi-step structure for the distribution function. The extreme value distribution for the circle map with an irrational parameter is obtained in closed form with the help of the three-gap theorem for the map X_j+1 = (X_j + a) mod 1 where a ∈ (0,1) is an irrational number.

The focus of this paper is on the distribution function of the rightmost eigenvalue for the complex elliptic Ginibre ensemble in the limit of weak non-Hermiticity. We show how the limiting distribution function can be expressed in terms of an integro-differential Painlevé-II function and how the same captures the nontrivial transition between Poisson and Airy point process extreme value statistics as the degree of non-Hermiticity decreases. Our most explicit new results concern the tail asymptotics of the limiting distribution function. For the right tail we compute the leading order asymptotics uniformly in the degree of non-Hermiticity, for the left tail we compute it close to Hermiticity.