Multilevel Dyson Brownian motions via Jack polynomials

Abstract

We introduce multilevel versions of Dyson Brownian motions of arbitrary parameter β > 0, generalizing the interlacing reflected Brownian motions of Warren for β = 2. Such processes unify β corners processes and Dyson Brownian motions in a single object. Our approach is based on the approximation by certain multilevel discrete Markov chains of independent interest, which are defined by means of Jack symmetric polynomials. In particular, this approach allows to show that the levels in a multilevel Dyson Brownian motion are intertwined (at least for β ≥ 1) and to give the corresponding link explicitly.

Interacting particle systems at the edge of multilevel Dyson Brownian motions

“…When β = 1, that is in the case of real symmetric random matrices, an asymptotic connection to TASEP of a similar type as above is known (see [Ss], [BFPS], [FSW] and also [PrSp], [BR]), but this case is much less understood conceptually. In addition, while many of the studied particle systems have far reaching generalizations (see [BC], [BG], [BP], [BP] and [GS2]), the interactions between particles are non-local for the range of parameters corresponding to general β random matrix models. Naively, one might conclude that there are no connections between general β random matrix models and interacting particle systems with local interactions.…”

Intertwinings for General $$\beta $$ β -Laguerre and $$\beta $$ β -Jacobi Processes

“…For a rigorous construction of the analogous coupled process in the case of Dyson Brownian motions with β > 2, see Sect. 4 of [13]. In fact, for certain values of the parameters, the construction of the process with the generator above can be reduced to the results of [13] and a more detailed account will appear as part of the author's Ph.D. thesis [2].…”

“…In fact, for certain values of the parameters, the construction of the process with the generator above can be reduced to the results of [13] and a more detailed account will appear as part of the author's Ph.D. thesis [2]. As just mentioned, such a coupling was constructed for Dyson Brownian motion with β > 2 in [13] and in [3] (see also [23]) for copies of general one-dimensional diffusion processes, which in particular includes the squared Bessel (this corresponds to the Laguerre process of this note) and Jacobi cases for β = 2, when the interaction, between the two levels, entirely consists of local hard reflection and the transition kernels are explicit. Given such 2-level couplings, one can then iterate to construct a multilevel process in a Gelfand-Tsetlin pattern, as in [25] which initiated this program (see also [13], [19], [3]).…”

“…As just mentioned, such a coupling was constructed for Dyson Brownian motion with β > 2 in [13] and in [3] (see also [23]) for copies of general one-dimensional diffusion processes, which in particular includes the squared Bessel (this corresponds to the Laguerre process of this note) and Jacobi cases for β = 2, when the interaction, between the two levels, entirely consists of local hard reflection and the transition kernels are explicit. Given such 2-level couplings, one can then iterate to construct a multilevel process in a Gelfand-Tsetlin pattern, as in [25] which initiated this program (see also [13], [19], [3]). For a different type of coupling, for β = 2 Dyson Brownian motion preceded [15] and is related to the Robinson-Schensted correspondence; see [16], [17] and the related work [5].…”

“…The aim of this short note is to establish intertwining relations between the semigroups of general β-Laguerre and β-Jacobi processes, in analogy to the ones obtained for general β-Dyson Brownian motion in [20] (see also [13]). These also generalize the relations obtained for β = 2 in [3] when the transition kernels for these semigroups are given explicitly in terms of h-transforms of Karlin-McGregor determinants.…”