The logarithmic derivative for point processes with equivalent Palm measures

Bufetov, Alexander I.; Dymov, Andrey

doi:10.2969/jmsj/78397839

Cited by 6 publications

(2 citation statements)

References 21 publications

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“…To show ISDEs by the Dirichlet form approach, expression of the logarithmic derivatives is crucial, because the logarithmic derivatives correspond the drift terms of ISDEs. Bufetov, Dymov, and Osada introduced a method to compute the logarithmic derivatives for determinantal random point fields [2]. Since μ α is determinantal, their result seems to be applicable to our case.…”

Section: Introductionmentioning

confidence: 89%

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Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices

Kawamoto¹

2022

Modern Stochastics: Theory and Applications

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The generalised sine random point field arises from the scaling limit at the origin of the eigenvalues of the generalised Gaussian ensembles. We solve an infinite-dimensional stochastic differential equation (ISDE) describing an infinite number of interacting Brownian particles which is reversible with respect to the generalised sine random point field. Moreover, finite particle approximation of the ISDE is shown, that is, a solution to the ISDE is approximated by solutions to finite-dimensional SDEs describing finite-particle systems related to the generalised Gaussian ensembles.

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Section: Introductionmentioning

confidence: 89%

“…Note that μ N G,α is a determinantal random point field. Let {p n α } n∈N be the monic orthogonal polynomials with respect to |x| 2α e −x 2 dx, and we set h n = R (p n α (x)) 2 × |x| 2α e −x 2 dx. Let K N G,α : R × R → R be the determinantal kernel defined as…”

Section: Introductionmentioning

confidence: 99%

Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices

Kawamoto¹

2022

Modern Stochastics: Theory and Applications

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Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions

Osada

2021

Sugaku Expositions

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We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.

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Stochastic analysis of infinite particle systems—A new development in classical stochastic analysis and dynamical universality of random matrices

Osada

2023

Sugaku Expositions

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We examine the stochastic dynamics of infinite particle systems moving in R d \mathbb {R}^{d} . The classical stochastic analysis pursues the stochastic dynamics of a single particle. We explain how to extend the classical stochastic analysis when the object becomes an infinite particle system. The equilibrium states of a system of infinite number of particles resulting from random matrices have the logarithmic function as an interaction potential. Hence, we develop a theory that applies to interaction potentials having a dominating influence over long distances. As an application, we show that the universality of point processes related to random matrices holds for stochastic dynamics.

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The logarithmic derivative for point processes with equivalent Palm measures

Cited by 6 publications

References 21 publications

Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices

Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices

Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions

Stochastic analysis of infinite particle systems—A new development in classical stochastic analysis and dynamical universality of random matrices

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