Determinantal Point Processes, Stochastic Log-Gases, and Beyond

Abstract

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures, whose correlation functions are all given by determinants specified by an integral kernel called the correlation kernel. First we show our new scheme of DPPs in which a notion of partial isometies between a pair of Hilbert spaces plays an important role. Many examples of DPPs in one-, two-, and higher-dimensional spaces are demonstrated, where several types of weak convergence from finite DPPs to infinite DPPs are given. Dynamical extensions of DPP are realized in one-dimensional systems of diffusive particles conditioned never to collide with each other. They are regarded as one-dimensional stochastic log-gases, or the two-dimensional Coulomb gases confined in one-dimensional spaces. In the second section, we consider such interacting particle systems in one dimension. We introduce a notion of determinantal martingale and prove that, if the system has determinantal martingale representation (DMR), then it is a determinantal stochastic process (DSP) in the sense that all spatio-temporal correlation function are expressed by a determinant. In the last section, we construct processes of Gaussian free fields (GFFs) on simply connected proper subdomains of $\mathbb{C}$ coupled with interacting particle systems defined on boundaries of the domains. There we use multiple Schramm--Loewner evolutions (SLEs) driven by the interacting particle systems. We prove that, if the driving processes are time-changes of the log-gases studied in the second section, then the obtained GFF with multiple SLEs are stationary. The stationarity defines an equivalence relation of GFFs, which will be regarded as a generalization of the imaginary surface studied by Miller and Sheffield.

arXiv commentv2: LaTeX 79 pages, no figure. This manuscript was prepared for the mini course given at `Workshop on Probability and Stochastic Processes' held at Orange County, Coorg, India, from 23rd to 26th February, 2020, which was organized by the Indian Academy of Sciences, Bangalore. arXiv admin note: substantial text overlap with arXiv:1903.04945

References in corpus (22)

• The Calogero-Sutherland Model and Generalized Classical Polynomials
• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• Universality and scaling of correlations between zeros on complex manifolds
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• Differential Equations for Dyson Processes
• Conditional Intensity and Gibbsianness of Determinantal Point Processes
• The ASEP and determinantal point processes
• On Multiple Schramm-Loewner Evolutions
• Zeros of Airy Function and Relaxation Process
• Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals
• Gaussian free field and conformal field theory
• Determinantal Martingales and Noncolliding Diffusion Processes
• Noncolliding processes, matrix-valued processes and determinantal processes
• Dynamical Correlations for Circular Ensembles of Random Matrices
• Dispersionless BKP Hierarchy and Quadrant Löwner Equation
• From Random Polynomials to Symplectic Geometry
• Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
• Conformal welding problem, flow line problem, and multiple Schramm--Loewner evolution
• On the Uniqueness of Global Multiple SLEs
• Partial Isometries, Duality, and Determinantal Point Processes