Directed random polymers via nested contour integrals

Borodin, Alexei; Bufetov, Alexander I.; Corwin, Ivan

doi:10.1016/j.aop.2016.02.001

Cited by 53 publications

(100 citation statements)

References 89 publications

(229 reference statements)

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“…In the totally asymmetric case (p D 1,˛> 0, q D D 0) [3] proved a KPZ universality class limit theorem with either square root / cube root fluctuations and Gaussian / GUE Tracy-Widom-type statistics (depending on the exact strength of˛). For the KPZ equation itself, [11,49] have employed the nonrigorous replica Bethe ansatz methods to derive a similar set of KPZ universality class limit theorems as was shown in the TASEP. In the partially asymmetric reflecting case (˛D D 0) Tracy-Widom derived explicit formulas for the configuration probabilities in [68], though no asymptotics have been accessible from these formulas as of yet (one generally expects from the universality belief that the same sort of dichotomy between square root and cube root fluctuations exists for the general partially asymmetric case.…”

Section: Existing Half-line Asep Resultsmentioning

confidence: 99%

Open ASEP in the Weakly Asymmetric Regime

Corwin

Shen

2018

Comm Pure Appl Math

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154

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We consider ASEP on a bounded interval and on a half-line with sources and sinks. On the full line, Bertini and Giacomin in 1997 proved convergence under weakly asymmetric scaling of the height function to the solution of the KPZ equation. We prove here that under similar weakly asymmetric scaling of the sources and sinks as well, the bounded interval ASEP height function converges to the KPZ equation on the unit interval with Neumann boundary conditions on both sides (different parameter for each side), and likewise for the half-line ASEP to KPZ on a half-line. This result can be interpreted as showing that the KPZ equation arises at the triple critical point (maximal current / high density / low density) of the open ASEP.

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Section: Existing Half-line Asep Resultsmentioning

confidence: 99%

Open ASEP in the Weakly Asymmetric Regime

Corwin

Shen

2018

Comm Pure Appl Math

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154

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“…There are further integrable generalizations of the LL model associated to reflection groups G of R n (the so-called generalized kaleidoscope, see section 5.2 of [104]). A large class can be indexed by root systems of simple Lie groups [80,106] and contain the half-space model studied in this paper, which has a tunable interaction parameter b with the wall at x = 0 [78,104,[107][108][109]. Here b is an arbitrary real number and the case b = +∞ corresponds to the hard wall boundary conditions.…”

Section: Overview: the Attractive Lieb-liniger Model On The Half-linementioning

confidence: 99%

“…The results for this case have been presented in a short form in [76]. The case b = 0 corresponds to the symmetric case and was studied in [78]. The droplet initial condition corresponds to the DP with fixed endpoints.…”

Section: The Kpz Equation and The Directed Polymer In A Half-spacementioning

confidence: 99%

“…Here we first consider the quantum mechanics of n ∈ N bosons on a line X = (x 1 , · · · , x n ) ∈ [0, L] n with attractive zero-range interaction. We use units such that The full space model then has initial condition Z(x, 0) = 2δ(x) instead of Z(x, 0) = δ(x) see [78]. = 1 and the mass of the particle is m = 1/2.…”

Section: Finite Size Modelmentioning

confidence: 99%

“…For b = +∞, which models a DP with an infinitely repulsive hard wall, it was obtained in [76] using the RBA method, and more recently in a distinct, although equivalent form in [77]. Another solution (also non-rigorous) was obtained for b = 0 in [78], with related but different methods using nested contour integral representations, initiated in [79] in physics and [80] in mathematics. In both cases the large time limit is found to be GSE-TW, consistent with the general picture of BR discussed above.…”

Section: Introduction and Aim Of The Papermentioning

confidence: 99%

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Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

Nardis¹,

Krajenbrink²,

Doussal³

et al. 2020

J. Stat. Mech.

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We revisit the Lieb-Liniger model for n bosons in one dimension with attractive delta interaction in a half-space R + with diagonal boundary conditions. This model is integrable for arbitrary value of b ∈ R, the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as b is decreased from the hard-wall case b = +∞, with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar-Parisi-Zhang equation for the growth of a one-dimensional interface of height h(x, t), on the half-space with boundary condition ∂ x h(x, t)| x=0 = b and droplet initial condition at the wall. We obtain explicit expressions, valid at all time t and arbitrary b, for the integer exponential (one-point) moments of the KPZ height field e nh(0,t) . From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for b > − 1 2 where the PDF is given by the GSE Tracy-Widom distribution (ii) bound for b < − 1 2 , where the PDF is a Gaussian. At the critical point b = − 1 2 , the PDF is given by the GOE Tracy-Widom distribution.Overview: KPZ in a half-space.There has been much recent progress in physics and mathematics in the study of the 1D (KPZ) universality class, thanks to the discovery of exact solutions and the development of powerful methods to address stochastic integrability and integrable probability. The KPZ class includes a host of models [1]: discrete versions of stochastic interface growth such as the PNG growth model [2][3][4], exclusion processes such as the TASEP, the ASEP, the q-TASEP and other variants [5][6][7][8][9][10][11][12][13], discrete (i.e. square lattice) [3,[14][15][16][17][18][19][20][21][22] or semi-discrete [23-25] models of directed polymers (DP) at zero and finite temperature, random walks in time dependent random media [26,27], dimer models, random tilings, random permutations [28], correlation function in quantum condensates [29][30][31], and more. At the center of this class lies the continuum KPZ equation [32], see equation (3), which describes the stochastic growth of a continuum interface, and its equivalent formulation in terms of continuum directed polymers (DP) [33], via the Cole-Hopf mapping onto the stochastic heat equation (SHE). Recently exact solutions have also been obtained for the KPZ equation at all times for various initial conditions [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. This was achieved by two different routes. First by studying scaling limits of solvable discrete models, which allowed for rigorous treatments. The second, pioneered by Kardar [51], is non-rigorous, but leads to a more direct solution: it starts from the DP formulation, uses the replica method together with a mapping to the attractive delta-Bose gas (LL model), whi...

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Two-Point Convergence of the Stochastic Six-Vertex Model to the Airy Process

Dimitrov

2023

Commun. Math. Phys.

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Directed random polymers via nested contour integrals

Cited by 53 publications

References 89 publications

Open ASEP in the Weakly Asymmetric Regime

Open ASEP in the Weakly Asymmetric Regime

Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

Two-Point Convergence of the Stochastic Six-Vertex Model to the Airy Process

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