Free-energy distribution of the directed polymer at high temperature

Abstract

We study the directed polymer of length t in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal regime of high temperature T is described, upon scaling 'time' t ∼ T 5 /κ and space x = T 3 /κ (with κ = T for the discrete model) by a continuum model with δ-function disorder correlation. Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. We then obtain the exact expression at any time for the generating function of the free energy distribution, in terms of a Fredholm determinant. At large time we find that it crosses over to the Tracy Widom distribution (TW) which describes the fixed T infinite t limit. The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models. PACS numbers:The directed polymer (DP) in a random potential provides the simplest example of a glass phase induced by quenched disorder [1] and has numerous applications, e.g. vortex lines [2], domain walls [3], biophysics [4]. It is closely related to much studied growth models in the KPZ class [5], such as asymmetric exclusion processes (ASEP) [6,7], and to Burgers turbulence [8]. It belongs to the broader class of disordered elastic manifolds, known to exhibit statistically scale invariant ground states. Within the functional RG (FRG) [9] these were described in a dimensional expansion by T = 0 fixed points, where the ratio temperature/disorder is irrelevant and scales with internal size with exponent −θ.Exact results were obtained in dimension d = 1 + 1 [1]. Johansson proved [10,11] that (i) the minimal energy path of length t on a square lattice with fixed endpoints has transverse roughness x ∼ t ζ with ζ = 2 3 (ii) the fluctuation of the ground state energy grows as t θ with θ = 1 3 and its scaled distribution coincides with the one of the smallest eigenvalue of a hermitian random matrix, the GUE Tracy Widom (TW) distribution [12]. The TW distribution was found in many other related models, polynuclear growth [13], TASEP [6], random subsequences [14,15] and others [16][17][18]. The unifying concept of determinantal space-time process and edge scaling was studied to account for such universality [19]. An exact result for the space-time scaling function of the two-point correlator of the height in KPZ was obtained [20].On the other hand, in d = 1 + 1 the model can be mapped onto the quantum mechanics of n attractive bosons in the limit n = 0, where t plays the role of (imaginary) time. It can be solved with the Bethe Ansatz (BA) for δ-function interactions. Until now only the ground state energy E 0 (n) was studied, i.e. the limit t → ∞ first. Pioneering attempts at its direct analytical continuation at n = 0 for a system of transverse size L = ∞ led to scaling behavior [21,22], but not to free energy...

Quantum Entanglement Growth under Random Unitary Dynamics

“…Finally, recent advances in KPZ theory have yielded an analytical expression for the full probability distribution of the KPZ height field [56][57][58][59][60][61][62][63][64][65][66]. In Appendix F we show that this analytical result compares well with Clifford numerics, providing further support for KPZ universality in this system.…”

“…The ratio C=B is universal (the constants v E and B are not). The KPZ fluctuations are non-Gaussian: remarkably, their universal probability distribution has been determined analytically [56][57][58][59][60][61][62][63][64][65][66]. The correlation length governing spatial correlations in the fluctuations grows with time as…”

“…As we mention in the main text, a remarkable recent advance in KPZ theory has been the derivation of the full universal probability distribution for the height of the surface at fixed position and fixed large time [56][57][58][59][60][61][62][63][64][65][66]; see Refs. [99][100][101] for reviews.…”

Short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation: Starting from a parabola

“…Several groups derived exact representations of this distribution [that we will call P(H, t, L)] for an arbitrary time t > 0. This remarkable progress has been achieved for three classes of initial conditions (and some of their combinations and variations): flat interface [9], sharp wedge [4,[10][11][12][13], and stationary interface: a two-sided Brownian interface pinned at a point [14,15]. In the long-time limit, and for typical fluctuations, P(H, t) converges to the Gaussian orthogonal ensemble (GOE) Tracy-Widom distribution [16] for the flat interface, to the Gaussian unitary ensemble (GUE) * Electronic address: kamenev@physics.umn.edu † Electronic address: meerson@mail.huji.ac.il ‡ Electronic address: pavel.sasorov@gmail.com Tracy-Widom distribution for the sharp wedge, and to the Baik-Rains distribution [17] for the stationary interface.…”

A KPZ Cocktail-Shaken, not Stirred...

“…This included the full time-evolution of the universal PDFs to their asymptotic TW forms, managed by independent researchers using complementary driven lattice-gas [70,71] and replica-theoretic DPRM approaches [72,73], in the former instance relying heavily upon recently gained insights into the weakly asymmetric simple exclusion process [74]. The initial advance was for the KPZ wedge geometry with its TW-GUE connection.…”