Extreme value statistics and some applications in statistical physics

Abstract

These notes are based on lectures delivered by G. Schehr at the XVIth School on Fundamental Problems in Statistical Physics (FPSP), held in Oropa (Italy) from 30 June to 11 July 2025. After a brief introduction to extreme value statistics (EVS) for independent and identically distributed (IID) random variables, we discuss several paradigmatic examples of strongly correlated systems where classical extreme value theory no longer applies. In particular, we focus on time series generated by random walks and Brownian motion, as well as on eigenvalue statistics in random matrix theory. Emphasis is placed on applications of EVS to fundamental problems in statistical physics and disordered systems, including the Random Energy Model, stochastic search problems, as well as fluctuating interfaces, and directed polymers in random media within the Kardar-Parisi-Zhang universality class.

Figures (9)

• Figure 1: An example of a time series, here, a set of IID variables with Gaussian distribution with zero mean and unit variance.
• Figure 2: Examples of disordered systems where the description of the extreme value statistics plays a key role in understanding their physical properties. In panel (a), a cartoon of a particle diffusion in a rough energy landscape (e.g., Sinai's model). Here, $z_{\max}$ and $z_{\min}$ denote respectively the positions of the maximum and the minimum of the potential $V(z)$. In panel (b), a representation of two configurations of a directed polymer on a square lattice. Note that they are correlated, since they share sites after the second, fifth, and eighth steps.
• Figure 3: Two physical interpretations of the CDF of $X_{\max}$. In panel (a), we provide a cartoon of $N$ particles on a line in the presence of the wall placed at $w$. Associating an energy to configurations of the particles' positions according to \ref{['eq:E_from_Pjoint']} renders the canonical partition function of the system equal to $Q_{\max}(w, N)$, as indicated in \ref{['eq:CDF_as_partition']}. In panel (b), we plot a trajectory of a Brownian motion. The survival probability of the trajectory staying below $w$ is equivalent to $Q_{\max}(w,N)$, with $N$ denoting the number of time steps.
• Figure 4: In panel (a), we plot the edge of a distribution $p(x) = 2/\pi \sqrt{1 - (x-1)^2}$ which has a support bounded at $X^* = 2$, and which vanishes as $p(x)\propto (X^*-x)^{1/2}$ as $x \to X^{*-}$. As we discuss further, such a parent distribution falls into the Weibull universality class with $\alpha = 3/2$, c.f., \ref{['eq:Wiebull_parent']}. In panel (b), we provide a cartoon illustrating that for $N$ large enough, there is almost surely $\mathcal{O}(1)$ of variables $x_i$ in the interval $[\mu_N, X^*]$.
• Figure 5: In panel (a), we plot the CDF of the maximum of a time series obtained for $N=2000$ IID variables distributed exponentially \ref{['eq:p_exp']} which fall into the Gumbel universality class as $N \to \infty$. Notice its sigmoidal shape, centered at $w = \ln N$ corresponding to the typical value of $X_{\max}$, obtained from \ref{['eq:typ_val_def']}. In panel (b), we plot the PDFs of the maximum of IID variables falling into the Gumbel, Fréchet and Weibull universality classes for $\rho$ = I, II, III respectively. The corresponding expressions for the limiting PDFs are given in Eqs. \ref{['eq:gumbel_dist']}, \ref{['eq:frechet_dist']} and \ref{['eq:weibull_dist']}. Note that the position of the bell-shaped curves is set by $a_N$, whereas their width is determined by $b_N$ -- see Eq. (\ref{['eq:scaled_var']}).