Free-energy distribution of the directed polymer at high temperature

By Pasquale Calabrese, P. Doussal, A. Rosso. 2010

294 citations recorded by the source corpus.

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∼T5/κ and space x=T3/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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d015a49ef3685ba48a765a17a442ba2c6507d67a

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Citations in the corpus

From duality to determinants for q-TASEP and ASEP (2012, methodology): “By diagonalizing the Bosonic Hamiltonian (the limit of (C)) via the Bethe Ansatz, [16, 12] both made initial attempts at computing the Laplace transform via the replica trick.”
The Kardar-Parisi-Zhang equation and universality class (2011, methodology): “At the same time, a very different (and mathematically non-rigorous) approach was pursued by two groups of physicists, Dotsenko [62] and Calabrese-Le Doussal-Rosso [41].”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, methodology): “Almost simultaneously, another approach based on the correspondence to the directed polymer problem and the use of the Bethe ansatz [92] was established by Calabrese et al.[51] and by Dotsenko [52].”
Macdonald processes (2011, methodology): “Soon after, Calabrese, Le Doussal and Rosso [26] and Dotsenko [38] derived the above Laplace transform through a non-rigorous replica approach.”
Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence (2012, methodology): “[14, 23] rely on the replica method [63], while those in Refs.”
Tropical Combinatorics and Whittaker functions (2011, methodology): “ wedge result done independently and in parallel). These results have been reproduced via highly non-rigorous polymer replica trick methods using the Bethe Ansatz for the attractive delta bose gas in [27, 38]. The work of [67] established the solvability of a semi-discrete directed polymer which can”
Directed polymers and the quantum Toda lattice (2009, unknown): “See also [12, 13, 14, 15, 45] for related recent developments.”
Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions (2010, background): “The probability distribution for the free energy of the cont inuum directed random polymer, as well as for the solution to the stochastic hea t equation and the KPZ equation has been a subject of interest for many years, wi th a large physics literature (see, for example, [9], [13], [20] and referen”
Stochastic six-vertex model (2014, methodology): “d for more complicated systems including Asymmetric Simple Exclusion Process (ASEP) [TW4], [BCS], q{TASEP [BC], [FV], and the solution to KPZ stochastic partial dierential equation [ACQ], [SS], [D], [CDR], [BCF]; see also [BG2], [BP], [Cor3] for reviews. The process Xb 1;b 2(t) has a limit to ASEP w”
Height Fluctuations for the Stationary KPZ Equation (2014, methodology): “la for a q-deformed Laplace transform of the one-point distribution. This provides an alternative to the methods of [79–81] as well as a rigorous regularization of the replica method used in [40] and [32]. HEIGHT FLUCTUATIONS FOR THE STATIONARY KPZ EQUATION 6 (5) [36] constructed a line ensemble ext”
A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, background): “While the ground-state energy and wavefunction of this quantum system have been known for a long time, the recent works [27, 42] have succeeded in summing over the full spectrum of excited states of the many-body Hamiltonian.”
Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport (2011, unknown): “2655 [195] Calabrese P, LeDoussal P, Rosso A.”