Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension $d=1,2,3$ via a Monte-Carlo procedure in the disorder

Abstract

In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, Körner, Katzgraber and Hartmann \cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-Carlo Markov chain in the disorder. In this paper, we combine their Monte-Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution $P_d(E_0)$ for the directed polymer in a random medium of dimension $d=1,2,3$. In $d=1$, we check the validity of the algorithm by a direct comparison with the exact result, namely the Tracy-Widom distribution. In dimensions $d=2$ and $d=3$, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order $P_d(E_0)\sim 10^{-22}$. Our results are in agreement with Zhang's argument, stating that the negative tail exponent $\eta (d)$ of the asymptotic behavior $\ln P_d(E_0)\sim -|E_0{|}^{\eta (d)}$ as $E_0\to -\infty$ is directly related to the fluctuation exponent $\theta (d)$ (which governs the fluctuations $\Delta E_0(L)\sim L^{\theta (d)}$ of the ground state energy $E_0$ for polymers of length $L$) via the simple formula $\eta (d)=1/(1-\theta (d))$. Along the paper, we comment on the similarities and differences with spin-glasses.

13 pages, 16 figures

References in corpus (8)

• Exact scaling functions for one-dimensional stationary KPZ growth
• Global fluctuations and Gumbel statistics
• Universal Statistics of the Critical Depinning Force of Elastic Systems in Random Media
• Universality-class dependence of energy distributions in spin glasses
• Probing tails of energy distributions using importance-sampling in the disorder with a guiding function
• Statistics of low energy excitations for the directed polymer in a $1+d$ random medium ($d=1,2,3$)
• Numerical study of the directed polymer in a 1+3 dimensional random medium
• Exact Enumeration of Ground States in the Sherrington-Kirkpatrick Spin Glass

Cited by in corpus (9)

• Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization models in dimensions d=2 and d=3
• Probing the large deviations of the Kardar-Parisi-Zhang equation at short time with an importance sampling of directed polymers in random media
• Universality class of replica symmetry breaking, scaling behavior, and the low-temperature fixed-point order function of the Sherrington-Kirkpatrick model
• Distribution of Lee-Yang zeros and Griffiths singularities in the $\pm J$ model of spin glasses
• Testing Error Correcting Codes by Multicanonical Sampling of Rare Events
• Kardar-Parisi-Zhang universality class in ($d+1$)-dimensions
• A Monte Carlo Algorithm for Sampling Rare Events: Application to a Search for the Griffiths Singularity
• Non-equilibrium dynamics of polymers and interfaces in random media : conjecture $\psi =d_s/2$ for the barrier exponent
• Scaling at the OTOC Wavefront: free versus chaotic models