Shuffling algorithm for boxed plane partitions

Borodin, Alexei; Gorin, Vadim

doi:10.1016/j.aim.2008.11.008

Cited by 37 publications

(51 citation statements)

References 27 publications

(51 reference statements)

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“…The special cases of such processes arise naturally in the study of two-dimensional statistical mechanics systems such as random stepped surfaces and various types of tilings (cf. [6,9,10,34]). …”

Section: Our Resultsmentioning

confidence: 99%

“…On the discrete level our construction of the multilevel stochastic dynamics is based on a procedure introduced by Diaconis and Fill [16], which has recently been used extensively in the study of Markov chains on interlacing particle configurations (see e.g. [6,7,[9][10][11]). The idea is to use commuting Markov operators and conditional independence to construct a multilevel Markov chain with given single level marginals.…”

Section: Our Methodsmentioning

confidence: 99%

“…Our next goal is to define a stochastic dynamics on GT (N ) . The construction we use is parallel to those of [6,7,[9][10][11]; it is based on an idea going back to [16], which allows to couple the dynamics of Young diagrams of different sizes. We start from the degenerate discrete time dynamics λ 0 (n) = ∅, n ∈ N 0 and construct the discrete time dynamics of λ 1 , λ 2 , .…”

Section: Proposition 225 the Process |Xmentioning

confidence: 99%

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Multilevel Dyson Brownian motions via Jack polynomials

Gorin

Shkolnikov

2014

Probab. Theory Relat. Fields

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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.

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Section: Our Resultsmentioning

confidence: 99%

Section: Our Methodsmentioning

confidence: 99%

Section: Proposition 225 the Process |Xmentioning

confidence: 99%

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Multilevel Dyson Brownian motions via Jack polynomials

Gorin

Shkolnikov

2014

Probab. Theory Relat. Fields

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“…C k (x, t). Thus the kernel writes K(x, n, t; y, n, t) = w t (x) 12) which is conjugate to the kernel…”

Section: )mentioning

confidence: 99%

“…A shuffling algorithm for lozenge tilings of the hexagon (also known as boxed plane partitions) has been constructed in [12] using the formalism developed in this paper, see [12] for details.…”

Section: More General Growth Modelsmentioning

confidence: 99%

Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

Borodin

Ferrari

2013

Commun. Math. Phys.

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222

548

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We construct a family of stochastic growth models in 2 + 1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1 + 1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models.The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece.(2) The one-point fluctuations of the height function in the curved part are asymptotically normal with variance of order ln(t) for time t ≫ 1. (3) There is a map of the (2 + 1)-dimensional space-time to the upper half-plane H such that on space-like submanifolds the multipoint fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on H.

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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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Shuffling algorithm for boxed plane partitions

Cited by 37 publications

References 27 publications

Multilevel Dyson Brownian motions via Jack polynomials

Multilevel Dyson Brownian motions via Jack polynomials

Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

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