Directed polymer in a random medium of dimension1+3: Multifractal properties at the localization-delocalization transition

Abstract

We consider the model of the directed polymer in a random medium of dimension 1+3 , and investigate its multifractal properties at the localization-delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyze the statistics of the position weights w{L}(r[over]) of the endpoint of the polymer of length L via the moments [equation: see text]. We measure the generalized exponents tau(q) and tau[over](q) governing the decay of the typical values [equation: see text] and disorder-averaged values Y{q}(L)[over] approximately L{-tau[over](q)} , respectively. To understand the difference between these exponents, tau(q) not equal to tau[over](q) above some threshold q>q{c} approximately 2 , we compute the probability distributions of [equation: see text] over the samples: We find that these distributions becomes scale invariant with a power-law tail 1/y{1+x{q}} . These results thus correspond to the Evers-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of inverse participation ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent nu approximately 2 .