Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights

O’Connell, Neil; Ortmann, Janosch

doi:10.1214/ejp.v20-3787

Cited by 43 publications

(72 citation statements)

References 14 publications

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“…The Tracy-Widom limit of the strict-weak polymer model was proved independently and concurrently by O'Connell and Ortmann [14]. They derived the Fredholm determinant formula (our Theorem 1.7) in a different way that complements our work.…”

Section: Introduction and Resultsmentioning

confidence: 56%

“…The pure alpha specialized q-Whittaker process converges [4,6] (under scaling related to those above) to the alpha specialized Whittaker process of [10]. As explained in [14], the analog of the smallest part for the pure alpha Whittaker process is related to the strict-weak polymer free energy. Methods coming from Whittaker processes [10] provide a route to write down a Laplace transform formula for the strict-weak polymer partition function which can be turned (using identities similar to those of [9]) into the Fredholm determinant formula present herein.…”

Section: Theorem 13mentioning

confidence: 88%

“…Methods coming from Whittaker processes [10] provide a route to write down a Laplace transform formula for the strict-weak polymer partition function which can be turned (using identities similar to those of [9]) into the Fredholm determinant formula present herein. This is the approach taken in [14]. We do not rely upon the connection to these Macdonald/q-Whittaker/Whittaker processes in the approach we utilize here, though certainly this was an important motivation in our pursuit.…”

Section: Theorem 13mentioning

confidence: 99%

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The Strict-Weak Lattice Polymer

2015

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We introduce the strict-weak polymer model, and show the KPZ universality of the free energy fluctuation of this model for a certain range of parameters. Our proof relies on the observation that the discrete time geometric q-TASEP model, studied earlier by Borodin and Corwin, scales to this polymer model in the limit q → 1. This allows us to exploit the exact results for geometric q-TASEP to derive a Fredholm determinant formula for the strictweak polymer, and in turn perform rigorous asymptotic analysis to show KPZ scaling and GUE Tracy-Widom limit for the free energy fluctuations. We also derive moments formulae for the polymer partition function directly by Bethe ansatz, and identify the limit of the free energy using a stationary version of the polymer model.

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Section: Introduction and Resultsmentioning

confidence: 56%

Section: Theorem 13mentioning

confidence: 88%

Section: Theorem 13mentioning

confidence: 99%

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The Strict-Weak Lattice Polymer

2015

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“…as well as the boundary condition (20). The condition (21) is enforced by asking that the rapidities satisfy the following Bethe equations ¶…”

Section: Bethe Ansatz Solution: Wave-functions and Bethe Equationsmentioning

confidence: 99%

“…formula and then comment it. On the space of symmetric functions of n variables (x 1 , · · · , x n ) ∈ [0, +∞[ n with the appropriate boundary condition (20)…”

Section: Decomposition Of the Identity In Terms Of Lieb-liniger Eigenmentioning

confidence: 99%

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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Stochastic Higher Spin Vertex Models on the Line

Corwin

Petrov

2015

Commun. Math. Phys.

129

226

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We introduce a four-parameter family of interacting particle systems on the line which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain Markov dualities. Using this, for the systems started in step initial data we write down nested contour integral formulas for moments and Fredholm determinant formulas for Laplace-type transforms. Taking various choices or limits of parameters, this family degenerates to many of the known exactly solvable models in the Kardar-Parisi-Zhang universality class, as well as leads to many new examples of such models. In particular, ASEP, the stochastic six-vertex model, q-TASEP and various directed polymer models all arise in this manner. Our systems are constructed from stochastic versions of the R-matrix related to the six-vertex model. One of the key tools used here is the fusion of R-matrices and we provide a probabilistic proof of this procedure.

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Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights

Cited by 43 publications

References 14 publications

The Strict-Weak Lattice Polymer

The Strict-Weak Lattice Polymer

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

Stochastic Higher Spin Vertex Models on the Line

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