Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers

By V. Dotsenko. 2010

238 citations recorded by the source corpus.

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 a many-body quantum boson system with attractive interactions. Performing the summation over the entire spectrum of excited states the problem is reduced to the Fredholm determinant with the Airy kernel which is known to yield the Tracy-Widom distribution.

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

View Paper: https://www.semanticscholar.org/paper/f4de16bc72fbf1a3c54b3b1174d7135f0bb95d1a

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Citations in the corpus

A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, background): “While the ground-state energy and wave function of this quantum system 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.”
Lectures on integrable probability (2012, methodology): “ of an analytic continuation argument for the functionwithspeciﬁedvaluesatintegerpoints. However,theﬁrstanswerstothisproblemobtainedviathereplicatrickwere incorrect. (Theywerelatercorrectedthough,cf. [Dotsnenko-10], [Calabrese-Doussal-Rosso-10].) A correct mathematical computation of theLaplacetrans”
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].”
Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport (2011, unknown): “The distribution of the height of the surface at 1388 any finite time is now known exactly (starting from some special initial conditions), 1389 solving a problem that remained open for 25 years: a description of the historical 1390 development of these results and the contributions of the various g”
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”
Extreme value statistics of correlated random variables: A pedagogical review (2019, unknown): “[198] V.”
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.”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, result): “Although in general the conversion from cumulants to distribution is not unique, and even worse in the present problem a rapidly diverging term appears, which cannot be treated with mathematical rigor, this approach successfully gave the GUE Tracy-Widom distribution for the narrow-wedge initial cond”
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 resetting and applications (2019, background): “Here, it is known that for times Tmicro t T ∗, and for a flat initial profile, the interface height H(x, t) has a deterministic linear growth with stochastic t fluctuations [129–137]: H(x, t) = v∞t+ (Γt) χ(x) .”
From duality to determinants for q-TASEP and ASEP (2012, methodology): “By diagonalizing the Bosonic Hamiltonian [the limit of (C)] via the Bethe ansatz, [12, 15] both made initial attempts at computing the Laplace transform via the replica trick.”
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.”
Stochastic six-vertex model (2014, methodology): “tained 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 A”