On a gateway between the Laguerre process and dynamics on partitions

Assiotis, Theodoros

doi:10.30757/alea.v16-38

Cited by 5 publications

(4 citation statements)

References 52 publications

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“…. , [4] (and hence we have weak continuity in the initial condition, see for example Theorem 2.5 in Chapter 4 of [29]). Towards this end, observe that using the semigroup property and by noting the explicit form (1), (3) of the transition kernels, we can write:…”

Section: Interacting Particle System At the Right Edgementioning

confidence: 99%

On the singular values of complex matrix Brownian motion with a matrix drift

Assiotis¹

2021

Preprint

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Let Mat C (K, N) be the space of K × N complex matrices. Let B t be Brownian motion on Mat C (K, N) starting from the zero matrix and M ∈ Mat C (K, N). We prove that, with K ≥ N, the N eigenvalues of (B t + tM) * (B t + tM) form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman [69] for multidimensional Brownian motion with drift which corresponds to N = 1. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe [75] and studied by Pitman and Yor [65], conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.

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Section: Interacting Particle System At the Right Edgementioning

confidence: 99%

On the singular values of complex matrix Brownian motion with a matrix drift

Assiotis¹

2021

Preprint

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“…Remark 2.8. We note that the recursive equations ( 4), (5) above and in particular their marginals on the left and right edge ( 6), (7), (8), (9) below are reminiscent but different 3 to the recursive equations that come up in the Robinson-Schensted-Knuth (RSK) [66] correspondence. This combinatorial algorithm can also be used to study certain first/last passage percolation models in inhomogeneous space in a dynamical way in terms of interacting particles, see [110,108,107,58,86,131,18].…”

Section: Recursive Equations For Discrete-time Dynamicsmentioning

confidence: 99%

“…We also introduce in Section 4.2 a more general setup for intertwinings of kernels that are given in terms of determinants and explain how our previous result fits into this framework. For some other intertwining relations that involve determinants, appearing in a different context, see for example [7,104].…”

Section: Relation To Previous Work Ideas and Techniquesmentioning

confidence: 99%

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

Assiotis

2020

Ann. Henri Poincaré

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We introduce a space inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari in [7]. We show that for a certain class of initial conditions the point process associated to the dynamics has determinantal correlation functions and we calculate explicitly, in the form of a double contour integral, the correlation kernel for one of the most classical initial conditions, the densely packed. En route to proving this we obtain some results of independent interest on non-intersecting general pure-birth chains, that generalize the Charlier process, the discrete analogue of Dyson's Brownian motion. Finally, these dynamics provide a coupling between the inhomogeneous versions of the TAZRP and PushTASEP particle systems which appear as projections on the left and right edges of the array respectively⋆ ⋆ + 1 ⋆ + 2 ⋆ + 3 ⋆ + 4 ⋆ + 5 ⋆ + 6 Figure 1: A configuration of particles in GT 4 . If the clock of the particle labelled X 3,1 4 rings, which happens at rate λ(⋆ + 1), then the move is blocked since interlacing with X 2,1 4 would be violated. On the other hand, if the clock of the particle X 2,2 4 rings, which happens at rate λ(⋆ + 3), then it jumps to the right by one and instantaneously pushes both X 3,3 4 and X 4,4 4 to the right by one as well, for otherwise the interlacing would break.The right edge particle system is called inhomogeneous PushTASEP, see [7], [6] and also [16] for a q-deformation of the homogeneous case. The fact that this particle system has an underlying determinantal structure is new as well, but is also obtained in the independent work of Petrov [46] that uses different methods.

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“…The term duality is also used in a fast growing and fascinating literature on this topic related to differential operators arising in statistical mechanics, see e.g. [3,12,18,33,19] and references therein. More generally, the concept of intertwining relation goes back to Dynkin [16] who used it to construct new Markov semigroups from a reference one.…”

Section: Introductionmentioning

confidence: 99%

Discrete self-similar and ergodic Markov chains

Miclo,

Patie,

Sarkar

2022

Preprint

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The first aim of this paper is to introduce a class of Markov chains on Z+ which are discrete self-similar in the sense that their semigroups satisfy an invariance property expressed in terms of a discrete random dilation operator. After showing that this latter property requires the chains to be upward skip-free, we first establish a gateway relation, a concept introduced in [26], between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on R+. As a by-product, we prove that each of these Markov chains, after an appropriate scaling, converge in the Skorohod metric, to the associated self-similar Markov process. By a linear perturbation of the generator of these Markov chains, we obtain a class of ergodic Markov chains, which are non-reversible. By means of intertwining and interweaving relations, where the latter was recently introduced in [27], we derive several deep analytical properties of such ergodic chains including the description of the spectrum, the spectral expansion of their semigroups, the study of their convergence to equilibrium in the Φ-entropy sense as well as their hypercontractivity property.

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On a gateway between the Laguerre process and dynamics on partitions

Cited by 5 publications

References 52 publications

On the singular values of complex matrix Brownian motion with a matrix drift

On the singular values of complex matrix Brownian motion with a matrix drift

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

Discrete self-similar and ergodic Markov chains

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