Hydrodynamic Limit Equation for a Lozenge Tiling Glauber Dynamics

Laslier, Benoît; Toninelli, Fabio Lucio

doi:10.1007/s00023-016-0548-8

Cited by 5 publications

(16 citation statements)

References 21 publications

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“…One exception is the extension to this situation of a special model among the ones discussed in section 7. (A similar restriction on the rates appears also for lozenge tiling Glauber dynamics [28].) Given i ∈ Λ N we define…”

Section: Sp With Relaxed Exclusionmentioning

confidence: 94%

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Gradient structure and transport coefficients for strong particles

Gabrielli¹,

Krapivsky²

2018

J. Stat. Mech.

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We introduce and study a simple and natural class of solvable stochastic lattice gases. This is the class of Strong Particles. The name is due to the fact that when they try to jump to an occupied site they succeed pushing away a pile of particles. For this class of models we explicitly compute the transport coefficients. We also discuss some generalizations and the relations with other classes of solvable models.

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Section: Sp With Relaxed Exclusionmentioning

confidence: 94%

“…For example, it would be very interesting to consider strong versions of models of the type studied in Ref. [28].…”

Section: Strong Hard Rodsmentioning

confidence: 99%

Gradient structure and transport coefficients for strong particles

Gabrielli¹,

Krapivsky²

2018

J. Stat. Mech.

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“…where V (ρ) = 1 π sin(πρ 1 ) sin(πρ 2 ) sin(π(1 − ρ 1 − ρ 2 )) . (6.15) Equations (6.13) and (6.14) are proven in [13,Prop. 14], as a consequence of a combinatorial identity discovered in [3].…”

Section: 23mentioning

confidence: 95%

“…7], and −Λ(·) is the so-called Lobachevsky function. Let us recall that, as well known and discussed for instance in [13,Sec. 3], σ(·) is C ∞ , strictly negative and strictly convex in T, and note that µ(·) is also C ∞ and strictly positive in T. Moreover, when ρ approaches ∂T the function σ(ρ) vanishes, the functions µ(ρ) and σ i,j (ρ) become singular and {σ i,j } i,j=1,2 loses the strict positive definiteness.…”

Section: Definition 23 (Transition Rates)mentioning

confidence: 99%

“…The results of [15,21] were used as a building block in [1,12] to prove that, under some conditions on the geometry of the domain, the mixing time of the single-flip Glauber dynamics is of order L 2+o (1) , like that of the long-jump dynamics. Finally, in [13] we discovered that, in contrast with the single-flip dynamics, the long-jump one satisfies certain identities, that allow to conjecture an explicit form for the hydrodynamic equation, cf. (2.8) or equivalently (2.10).…”

Section: Introductionmentioning

confidence: 99%

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Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation

Laslier

Toninelli

2018

Commun. Math. Phys.

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We study a reversible continuous-time Markov dynamics of a discrete (2+1)-dimensional interface. This can be alternatively viewed as a dynamics of lozenge tilings of the L × L torus, or as a conservative dynamics for a two-dimensional system of interlaced particles. The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. We consider a particular choice of the transition rates, originally proposed in [15]: in terms of interlaced particles, a particle jump of length n that preserves the interlacement constraints has rate 1/(2n). This dynamics presents special features: the average mutual volume between two interface configurations decreases with time [15] and a certain one-dimensional projection of the dynamics is described by the heat equation [21].In this work we prove a hydrodynamic limit: after a diffusive rescaling of time and space, the height function evolution tends as L → ∞ to the solution of a non-linear parabolic PDE. The initial profile is assumed to be C 2 differentiable and to contain no "frozen region". The explicit form of the PDE was recently conjectured [13] on the basis of local equilibrium considerations. In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg-Landau model [7,16], here the mobility coefficient turns out to be a non-trivial function of the interface slope.2010 Mathematics Subject Classification: 60K35, 82C20, 52C20

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Dimer model and holomorphic functions on t‐embeddings of planar graphs

Chelkak¹,

Laslier

Russkikh

2023

Proceedings of London Math Soc

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We introduce the framework of discrete holomorphic functions on t-embeddings of weighted bipartite planar graphs; t-embeddings also appeared under the name Coulomb gauges in a recent paper (Kenyon, Lam, Ramassamy, and Russkikh, Dimers and circle patterns, 2018). We argue that this framework is particularly relevant for the analysis of scaling limits of the height fluctuations in the corresponding dimer models. In particular, it unifies both Kenyon's interpretation of dimer observables as derivatives of harmonic functions on T-graphs and the notion of s-holomorphic functions originated in Smirnov's work on the critical Ising model. We develop an a priori regularity theory for such functions and provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field. We also discuss how several more standard discretizations of complex analysis fit this general framework.M S C 2 0 2 0 82B20, 30G25 (primary) Dmitry Chelkak, on leave from St.

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Hydrodynamic Limit Equation for a Lozenge Tiling Glauber Dynamics

Abstract: International audienc

Cited by 5 publications

References 21 publications

Gradient structure and transport coefficients for strong particles

Gradient structure and transport coefficients for strong particles

Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation

Dimer model and holomorphic functions on t‐embeddings of planar graphs

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