Directed polymer in a random medium of dimension 1+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 analyse the statistics of the position weights $w_L(\vec{r})$ of the end-point of the polymer of length $L$ via the moments $Y_q(L)={\sum }_{\vec{r}}[w_L(\vec{r}){]}^q$. We measure the generalized exponents $\tau (q)$ and $\tilde{\tau }(q)$ governing the decay of the typical values $Y_q^{typ}(L)=e^{\overline{\ln Y_q(L)}}\sim L^{-\tau (q)}$ and disorder-averaged values $\overline{Y_q(L)}\sim L^{-\tilde{\tau }(q)}$ respectively. To understand the difference between these exponents, $\tau (q)\ne \tilde{\tau }(q)$ above some threshold $q>q_c\sim 2$, we compute the probability distributions of $y=Y_q(L)/Y_q^{typ}(L)$ 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 Ever-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 $\nu \sim 2$.

10 pages, 15 figures

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• Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions
• Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices
• Symmetry relation for multifractal spectra at random critical points
• Pre-freezing of multifractal exponents in Random Energy Models with logarithmically correlated potential
• On the multifractal statistics of the local order parameter at random critical points : application to wetting transitions with disorder
• Level repulsion exponent $\beta$ for Many-Body Localization Transitions and for Anderson Localization Transitions via Dyson Brownian Motion
• Random cascade models of multifractality : real-space renormalization and travelling-waves
• Stiff directed lines in random media
• Localization transition of stiff directed lines in random media