TY - JOUR

T1 - Log-gamma directed polymer with one free end via coordinate Bethe Ansatz

AU - Grange, Pascal

N1 - Publisher Copyright:
© 2017 IOP Publishing Ltd and SISSA Medialab srl.

PY - 2017/7/11

Y1 - 2017/7/11

N2 - The discrete polymer model with random Boltzmann weights with homogeneous inverse gamma distribution, introduced by Seppäläinen, is studied in the case of a polymer with one fixed and one free end. The model with two fixed ends has been integrated by Thiery and Le Doussal, using coordinate Bethe Ansatz techniques and an analytic-continuation prescription. The probability distribution of the free energy has been obtained through the replica method, even though the moments of the partition sum do not exist at all orders due to the fat tail in the distribution of Boltzmann weights. To extend this approach to the polymer with one free end, we argue that the contribution to the partition sums in the thermodynamic limit is localised on parity-invariant string states. This situation is analogous to the case of the continuum polymer with one free end, related to the Kardar-Parisi-Zhang equation with flat boundary conditions and solved by Le Doussal and Calabrese. The expansion of the generating function of the partition sum in terms of numbers of strings can also be transposed to the log-gamma polymer model, with the induced Fredholm determinant structure. We derive the large-time limit of the rescaled cumulative distribution function, and relate it to the GOE Tracy-Widom distribution. The derivation is conjectural in the sense that it assumes completeness of a family of string states, and expressions of their norms, already useful in the fixed-end problem, and extends heuristically the order of moments of the partition sum to the complex plane.

AB - The discrete polymer model with random Boltzmann weights with homogeneous inverse gamma distribution, introduced by Seppäläinen, is studied in the case of a polymer with one fixed and one free end. The model with two fixed ends has been integrated by Thiery and Le Doussal, using coordinate Bethe Ansatz techniques and an analytic-continuation prescription. The probability distribution of the free energy has been obtained through the replica method, even though the moments of the partition sum do not exist at all orders due to the fat tail in the distribution of Boltzmann weights. To extend this approach to the polymer with one free end, we argue that the contribution to the partition sums in the thermodynamic limit is localised on parity-invariant string states. This situation is analogous to the case of the continuum polymer with one free end, related to the Kardar-Parisi-Zhang equation with flat boundary conditions and solved by Le Doussal and Calabrese. The expansion of the generating function of the partition sum in terms of numbers of strings can also be transposed to the log-gamma polymer model, with the induced Fredholm determinant structure. We derive the large-time limit of the rescaled cumulative distribution function, and relate it to the GOE Tracy-Widom distribution. The derivation is conjectural in the sense that it assumes completeness of a family of string states, and expressions of their norms, already useful in the fixed-end problem, and extends heuristically the order of moments of the partition sum to the complex plane.

KW - quantum integrability (Bethe Ansatz)

KW - solvable lattice models

UR - http://www.scopus.com/inward/record.url?scp=85026826357&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/aa7285

DO - 10.1088/1742-5468/aa7285

M3 - Article

AN - SCOPUS:85026826357

SN - 1742-5468

VL - 2017

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

IS - 7

M1 - 073102

ER -