The distribution function of the free energy fluctuations in one-dimensional directed polymers with $δ$-correlated random potential is studied by mapping the replicated problem to the $N$-particle quantum boson system with attractive interactions. We find the full set of eigenfunctions and eigenvalues of this many-body system and perform the summation over the entire spectrum of excited states. It is shown that in the thermodynamic limit the problem is reduced to the Fredholm determinant with the Airy kernel yielding the universal Tracy-Widom distribution, which is known to describe the statistical properties of the Gaussian unitary ensemble as well as many other statistical systems.
LPTMS, Université Paris Saclay, CNRS（巴黎萨克雷大学，法国国家科学研究中心）；Laboratoire de Physique de l’Ecole Normale Supérieure, CNRS, ENS and PSL Université, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎文理研究大学，索邦大学，巴黎西岱大学）；Racah Institute of Physics, Hebrew University of Jerusalem（耶路撒冷希伯来大学拉卡物理研究所）
Directed Polymer Transfer Matrices as a Unified Generator of Distinct One-Point Fluctuation Laws
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；School of Science, Constructor University（Constructor大学理学院）；Department of Physics, Massachusetts Institute of Technology（麻省理工学院物理系）
