Pfaffian Schur processes and last passage percolation in a half-quadrant

Abstract

We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy-Widom distributed, GOE Tracy-Widom distributed, or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy-Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit.

v5: this version corrects mistakes in the definition of the crossover kernel in Section 2.5. v4: minor typos corrected. v3: 58 pages, to appear in Annals of Probability. The initial submission arXiv:1606.00525v1 has been splitted in two parts. This paper contains only the results on last passage percolation. The results on the facilitated TASEP appear in arXiv:1707.01923

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Cited by in corpus (20)

• The free boundary Schur process and applications I
• Half-space Macdonald processes
• Half-space stationary Kardar-Parisi-Zhang equation
• Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
• From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies
• Stationary half-space last passage percolation
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• 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
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation
• Random walk on nonnegative integers in beta distributed random environment
• Approximating the stationary distribution of the ASEP with open boundaries
• KPZ fluctuations in finite volume
• Steady state of the KPZ equation on an interval and Liouville quantum mechanics
• Mixing times and cutoff for the TASEP in the high and low density phase
• Pinning, diffusive fluctuations, and Gaussian limits for half-space directed polymer models
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• The second class particle in the half-line open TASEP