Many stochastic processes in natural and man-made systems exhibit occasional extreme events which are rare in comparison to usual outcomes. Though infrequent, such events can still carry decisive weight. For example the fastest sperm may dominate the fertilization process. In describing such extreme outcomes, a key result of classical extreme value theory (EVT) is that the limiting maximum probability distribution function for the most extreme event observed over some interval generally converges to one of three classes of distributions – the Weibull, Gumbel or Fréchet distributions – depending on the large x behaviour of ψ(x), the probability density of the underlying random variable.

Recently studies have aimed to extend EVT to a wide range of different models all having a global confinement of the dynamics, which induces correlations among the random variables. It is a common trait shared in many renewal processes, mass transport models and long-range interacting spin models such as the truncated inverse distance squared Ising model. These three models describe numerous physical systems, including zero crossings of Brownian motion, arrival times at a detector, and interacting systems. Somewhat similarly to the classical ensembles of statistical physics—e.g., microcanonical ensembles with fixed energy, volume or number of particles and canonical ensembles at fixed temperature—different constraints also yield rich physical behaviours specific to the ensemble or model. For each model there are several classes of limiting laws in the thermodynamic limit when the global constraint diverges.

In a new paper, LML External Fellow Eli Barkai and colleagues present results relevant to two main issues related to EVT in the presence of global constraints. First, they provide a complete set of relations between constrained EVT and much simpler quantifiers of the underlying stochastic dynamics. These relations are exact and valid for any value of the global constraint, i.e., both close to and far from the thermodynamic limit. For renewal processes, for example, they report a simple, exact relation between EVT and the mean number of renewals. Second, they also exploit the exact relations and consider the thermodynamic limit, recapping known limiting laws and describing additional ones. For renewal processes, they find dual scaling, i.e., a theory which describes both types of limiting behaviour. When no moment of the waiting times exists, the theory describes typical events and rare events; when only the first moment of the waiting times exists, the theory describes the correction to Fréchet’s law and its large deviations.

The paper is available at https://faculty.biu.ac.il/~barkaie/MarcPRE2020.pdf

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