Extreme Value Statistics of Hierarchically Correlated Variables: Deviation from Gumbel Statistics and Anomalous Persistence

Abstract

We study analytically the distribution of the minimum of a set of hierarchically correlated random variables $E_1$, $E_2$, $...$, $E_N$ where $E_i$ represents the energy of the $i$-th path of a directed polymer on a Cayley tree. If the variables were uncorrelated, the minimum energy would have an asymptotic Gumbel distribution. We show that due to the hierarchical correlations, the forward tail of the distribution of the minimum energy becomes highly nnon universal, depends explicitly on the distribution of the bond energies $\epsilon$ and is generically different from the super-exponential forward tail of the Gumbel distribution. The consequence of these results to the persistence of hierarchically correlated random variables is discussed and the persistence is also shown to be generically anomalous.