Multiplicative functionals on ensembles of non-intersecting paths

Abstract

The purpose of this article is to develop a theory behind the occurrence of "path-integral" kernels in the study of extended determinantal point processes and non-intersecting line ensembles. Our first result shows how determinants involving such kernels arise naturally in studying ratios of partition functions and expectations of multiplicative functionals for ensembles of non-intersecting paths on weighted graphs. Our second result shows how Fredholm determinants with extended kernels (as arise in the study of extended determinantal point processes such as the Airy_2 process) are equal to Fredholm determinants with path-integral kernels. We also show how the second result applies to a number of examples including the stationary (GUE) Dyson Brownian motion, the Airy_2 process, the Pearcey process, the Airy_1 and Airy_{2->1} processes, and Markov processes on partitions related to the z-measures.

32 pages, 1 figure

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Cited by in corpus (11)

• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• How flat is flat in random interface growth?
• Brownian Motions with One-Sided Collisions: The Stationary Case
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• TASEP and generalizations: Method for exact solution
• Intermediate disorder directed polymers and the multi-layer extension of the stochastic heat equation
• Extreme statistics of non-intersecting Brownian paths
• The hard-edge tacnode process for Brownian motion
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• TASEP fluctuations with soft-shock initial data
• Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains