Extreme-value statistics of hierarchically correlated variables deviation from Gumbel statistics and anomalous persistence - art. no. 046121

We study analytically the distribution of the minimum of a set of hierarchi cally correlated random variables E-1, E-2,..., E-N where E-i represents th e energy of the ith path of a directed polymer on a Cayley tree. If the var iables were uncorrelated, the minimum energy would have an asymptotic Gumbe l distribution. We show that due to the hierarchical correlations, the forw ard tail of the distribution of the. minimum energy becomes highly nonunive rsal, depends explicitly on the distribution of the bond energies epsilon, and is generically different from the superexponential forward tail of the Gumbel distribution. The consequence of these results to the persistence of hierarchically correlated random variables is discussed and the persistenc e is also shown to be generically anomalous.