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Berestycki N, Lis M, Qian W. Free boundary dimers: random walk representation and scaling limit. Probab Theory Relat Fields 2023; 186:735-812. [PMID: 37334240 PMCID: PMC10271954 DOI: 10.1007/s00440-023-01203-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2021] [Revised: 01/11/2023] [Accepted: 03/30/2023] [Indexed: 06/20/2023]
Abstract
We study the dimer model on subgraphs of the square lattice in which vertices on a prescribed part of the boundary (the free boundary) are possibly unmatched. Each such unmatched vertex is called a monomer and contributes a fixed multiplicative weight z > 0 to the total weight of the configuration. A bijection described by Giuliani et al. (J Stat Phys 163(2):211-238, 2016) relates this model to a standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this dimer model describes a walk with transition weights that are negative along the free boundary. Yet under certain assumptions, which are in particular satisfied in the infinite volume limit in the upper half-plane, we prove an effective, true random walk representation for the inverse Kasteleyn matrix. In this case we further show that, independently of the value of z > 0 , the scaling limit of the centered height function is the Gaussian free field with Neumann (or free) boundary conditions. It is the first example of a discrete model where such boundary conditions arise in the continuum scaling limit.
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Affiliation(s)
| | - Marcin Lis
- Technische Universität Wien, Vienna, Austria
| | - Wei Qian
- City University of Hong Kong, Kowloon Tong, Hong Kong
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Gaussian unitary ensemble in random lozenge tilings. Probab Theory Relat Fields 2022. [DOI: 10.1007/s00440-022-01168-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Beffara V, Chhita S, Johansson K. Local geometry of the rough-smooth interface in the two-periodic Aztec diamond. ANN APPL PROBAB 2022. [DOI: 10.1214/21-aap1701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | - Sunil Chhita
- Department of Mathematical Sciences, Upper Mountjoy Campus, Durham University
| | - Kurt Johansson
- Department of Mathematics, Royal Institute of Technology (KTH)
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Ferrari PL, Vető B. Fluctuations of the arctic curve in the tilings of the Aztec diamond on restricted domains. ANN APPL PROBAB 2021. [DOI: 10.1214/20-aap1590] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | - Bálint Vető
- Department of Stochastics, Budapest University of Technology and Economics; MTA – BME Stochastics Research Group
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Laslier B. Local limits of lozenge tilings are stable under bounded boundary height perturbations. Probab Theory Relat Fields 2018. [DOI: 10.1007/s00440-018-0853-x] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/14/2022]
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Giuliani A, Mastropietro V, Toninelli FL. Height fluctuations in interacting dimers. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2017. [DOI: 10.1214/15-aihp710] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Chhita S, Johansson K, Young B. Asymptotic domino statistics in the Aztec diamond. ANN APPL PROBAB 2015. [DOI: 10.1214/14-aap1021] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Adler M, Chhita S, Johansson K, van Moerbeke P. Tacnode GUE-minor processes and double Aztec diamonds. Probab Theory Relat Fields 2014. [DOI: 10.1007/s00440-014-0573-9] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Li Z. Uniqueness of the infinite homogeneous cluster in the 1-2 model. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2014. [DOI: 10.1214/ecp.v19-3105] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Falco P. Interacting fermions picture for dimer models. Phys Rev E 2013; 87:060101. [PMID: 23848610 DOI: 10.1103/physreve.87.060101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/25/2012] [Revised: 04/24/2013] [Indexed: 11/07/2022]
Abstract
Recent numerical results on classical dimers with weak aligning interactions have been theoretically justified via a Coulomb gas representation of the height random variable. Here, we propose a completely different representation, the interacting fermions picture, which avoids some difficulties of the Coulomb gas approach and provides a better account of the numerical findings. Besides, we observe that the Peierls argument explains the behavior of the system in the strong interaction case.
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Affiliation(s)
- P Falco
- Department of Mathematics, California State University, Northridge, California 91330, USA
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Abstract
Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details.
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Affiliation(s)
- Michael Baake
- Fakultät für Mathematik, Universität Bielefeld, Germany.
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Boutillier C, de Tilière B. The critical Z-invariant Ising model via dimers: the periodic case. Probab Theory Relat Fields 2009. [DOI: 10.1007/s00440-009-0210-1] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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Abstract
Fisher and Stephenson conjectured in 1963 that the correlation function (defined by dimer packings) of two unit holes on the square lattice is rotationally invariant in the limit of large separation between the holes. We consider the same problem on the hexagonal lattice, extend it to an arbitrary finite collection of holes, and present an explicit conjectural answer. In recent work we managed to prove this conjecture in two fairly general cases. The quantity giving the answer can be regarded as the exponential of the negative of the two-dimensional electrostatic energy of a system of charges naturally associated with the holes. We further develop this analogy to electrostatics by presenting two different natural ways to define a field in our setup, and showing that both lead to the electric field, in the limit of large separations between the holes. For one of the fields, this is also stated as a limit shape theorem for random surfaces, with the continuum limit being a sum of helicoids. We conclude by explaining the relationship of our results to previous results in the physics literature on spin correlations in the Ising model.
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Johansson K. Course 1 Random matrices and determinantal processes. MATHEMATICAL STATISTICAL PHYSICS, ÉCOLE D'ÉTÉ DE PHYSIQUE DES HOUCHES SESSION LXXXIII 2006. [DOI: 10.1016/s0924-8099(06)80038-7] [Citation(s) in RCA: 68] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/05/2023]
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Wilson DB. Red-green-blue model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:037105. [PMID: 15089445 DOI: 10.1103/physreve.69.037105] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2002] [Indexed: 05/24/2023]
Abstract
We experimentally study the red-green-blue model, which is a system of loops obtained by superimposing three dimer coverings on offset hexagonal lattices. We find that when the boundary conditions are "flat," the red-green-blue loops are closely related to stochastic Loewner evolution with parameter kappa=4 (SLE4) and double-dimer loops, which are the loops formed by superimposing two dimer coverings of the Cartesian lattice. But we also find that the red-green-blue loops are more tightly nested than the double-dimer loops. We also investigate the two-dimensional minimum spanning tree, and find that it is not conformally invariant.
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Affiliation(s)
- David B Wilson
- Microsoft Research, One Microsoft Way, Redmond, Washington 98052, USA
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Feigenbaum J, Ishai Y, Malkin T, Nissim K, Strauss MJ, Wright RN. Secure Multiparty Computation of Approximations. AUTOMATA, LANGUAGES AND PROGRAMMING 2001. [DOI: 10.1007/3-540-48224-5_75] [Citation(s) in RCA: 42] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/11/2022]
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