Crossover in the log-gamma polymer from the replica coordinate Bethe\n Ansatz

2017/07/11 by Pascal Grange, Grange, Pascal
Mathematics · Physics and Astronomy · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Random Matrices and Applications #Statistical Distribution Estimation and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1707.03521

openalex publication_date 2017/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The coordinate Bethe Ansatz solution of the log-gamma polymer is extended to\nboundary conditions with one fixed end and the other attached to one half of a\none-dimensional lattice. The large-time limit is studied using a saddle-point\napproximation,and the cumulative distribution function of the rescaled free\nenergy of a long polymer is expressed as a Fredholm determinant. Scaling limits\nof the kernel are identified, leading to a crossover from the GUE to the GOE\nTracy--Widom distributions. The continuum limit reproduces the crossover from\ndroplet to flat initial conditions of the Kardar--Parisi--Zhang equation.\n

Related

• Fluctuations of the log-gamma polymer free energy with general parameters and slopes
• Convergence from the Log-Gamma Polymer to the Directed Landscape
• An identity in distribution between full-space and half-space log-gamma polymers
• Maximal free energy of the log-gamma polymer
• The half-space log-Gamma polymer in the bound phase