Directed random polymers via nested contour integrals

Abstract

We study the partition function of two versions of the continuum directed polymer in 1+1 dimension. In the full-space version, the polymer starts at the origin and is free to move transversally in the reals, and in the half-space version, the polymer starts at the origin but is reflected at the origin and stays in the negative reals. The partition functions solves the stochastic heat equation in full-space or half-space with mixed boundary condition at the origin; or equivalently the free energy satisfies the Kardar-Parisi-Zhang equation. We derive exact formulas for the Laplace transforms of the partition functions. In the full-space this is expressed as a Fredholm determinant while in the half-space this is expressed as a Fredholm Pfaffian. Taking long-time asymptotics we show that the limiting free energy fluctuations scale with exponent 1/3 and are given by the GUE and GSE Tracy-Widom distributions. These formulas come from summing divergent moment generating functions, hence are not mathematically justified. The primary purpose of this work is to present a mathematical perspective on the polymer replica method which is used to derive these results. In contrast to other replica method work, we do not appeal directly to the Bethe ansatz for the Lieb-Liniger model but rather utilize nested contour integral formulas for moments as well as their residue expansions.

62 pages, 2 figures

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• Half-space Macdonald processes
• Large fluctuations of the KPZ equation in a half-space
• Half-space stationary Kardar-Parisi-Zhang equation
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line
• Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
• Probing the large deviations of the Kardar-Parisi-Zhang equation at short time with an importance sampling of directed polymers in random media
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• When fast and slow interfaces grow together: connection to the half-space problem of the Kardar-Parisi-Zhang class
• Circular Kardar-Parisi-Zhang interfaces evolving out of the plane
• The lower tail of the half-space KPZ equation
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Lyapunov exponents of the half-line SHE
• Stationary measures of the KPZ equation on an interval from Enaud-Derrida's matrix product ansatz representation
• Boundary current fluctuations for the half space ASEP and six vertex model
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• Random walk on nonnegative integers in beta distributed random environment
• Steady state of the KPZ equation on an interval and Liouville quantum mechanics
• Height distributions in interface growth: The role of the averaging process