Universal and non-universal tails of distribution functions in the directed polymer and KPZ problems

Abstract

The optimal fluctuation approach is applied to study the most distant (non-universal) tails of the free-energy distribution function P(F) for an elastic string (of a large but finite length L) interacting with a quenched random potential. A further modification of this approach is proposed which takes into account the renormalization effects and allows one to study the most close (universal) parts of the tails. The problem is analyzed for different dimensions of a space in which the polymer is imbedded. In terms the stochastic growth problem, the same distribution function describes the distribution of heights in the regime of a non-stationary growth in a situation when an interface starts to grow from a flat configuration.

arXiv comment17 pages, 2 figures, the final version, two paragraphs added to the conclusion

References in corpus (14)

• Shape Fluctuations and Random Matrices
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Exact scaling functions for one-dimensional stationary KPZ growth
• Instantons in Burgers Equation
• Intermittency of Burgers' Turbulence
• Viscous Instanton for Burgers' Turbulence
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• Large negative velocity gradients in Burgers turbulence
• Optimal fluctuation approach to a directed polymer in a random medium
• Correlations, soliton modes, and non-Hermitian linear mode transmutation in the 1D noisy Burgers equation
• Directed polymers in a random medium: a variational approach
• Joint free energy distribution in the random directed polymer problem
• The Wandering Exponent of a One-Dimensional Directed Polymer in a Random Potential with Finite Correlation Radius
• Directed polymers and interfaces in random media : free-energy optimization via confinement in a wandering tube

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