Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz

Abstract

We show how our previous result based on the replica Bethe ansatz for the Kardar Parisi Zhang (KPZ) equation with the "half-flat" initial condition leads to the Airy${}_2$ to Airy${}_1$ (i.e. GUE to GOE) universal crossover one-point height distribution in the limit of large time. Equivalently, we obtain the distribution of the free energy of a long directed polymer (DP) in a random potential with one fixed endpoint and the other one on a half-line. We then generalize to a DP when each endpoint is free on its own half-line. It amounts, in the limit of large time, to obtain the distribution of the maximum of the transition process Airy${}_{2\to 1}$ (minus a half-parabola) on a half line.

27 pages, 3 figures

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• From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation
• On integrable directed polymer models on the square lattice
• How flat is flat in random interface growth?
• Topology and the Kardar-Parisi-Zhang universality class
• Exact formulas for random growth with half-flat initial data
• Tail of the two-time height distribution for KPZ growth in one dimension
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• Log-Gamma directed polymer with fixed endpoints via the replica Bethe Ansatz
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• Crossover between various initial conditions in KPZ growth: flat to stationary
• Log-gamma directed polymer with one free end via coordinate Bethe Ansatz
• Distribution of Brownian coincidences
• Analytical Methods and Field Theory for Disordered Systems
• Crossover in the log-gamma polymer from the replica coordinate Bethe Ansatz