Non-intersecting paths, random tilings and random matrices

By K. Johansson. 2000

296 citations recorded by the source corpus.

Abstract. We investigate certain measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact can be obtained from non-intersecting Brownian motions. The derivations of the measures are based on the Karlin-McGregor or Lindström-Gessel-Viennot method. We use the measures to show some asymptotic results for the models.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Determinantal Point Processes for Machine Learning (2012, background): “Recently, DPPs have attracted a flurry of attention in the mathematics community [13, 14, 15, 16, 21, 72, 73, 74, 116, 117, 131], and much progress has been made in understanding their formal combinatorial and probabilistic properties.”
Zeros of Gaussian Analytic Functions and Determinantal Point Processes (2009, background): “However the Hermitian assumption is not essential and indeed, there are interesting examples of determinantal processes with non-Hermitian kernels (particularly those that involve “time” or “dynamics”, see (41) for some examples).”
Determinantal probability measures (2002, methodology): “Introduction 3 instances have been studied in Borodin (2000), Borodin and Olshanski (2000, 2001, 2002), Borodin, Okounkov, and Olshanski (2000), Johansson (2001, 2002), Okounkov (2001), and Okounkov and Reshetikhin (2001).”
The Surprising Mathematics of Longest Increasing Subsequences (2015, background): “Two other arctic circle theorems are proved in [28] (see also [64]) and [98].”
Random matrix theory (2019, background): “Most of the work on the topics in this section has been done by Johansson and his co-authors, and the best place to find out more is to read his papers, for example, [2], [14], [15], [16].”
A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, background): “For further information we refer the reader to the surveys [1, 60, 68, 88, 125, 129] and the monograph [11].”
Lectures on integrable probability (2012, background): “s are known including (in addition to the random matrix theory) dimers on bipartite graphs [Kenyon-09], uniform spanning trees [Lyons-03], ensembles of non-intersecting paths on planar acyclic graphs [Johansson-02] and zeros of random analytic functions [Hough-Krishnapur-Virag-Peres-10]. A recent re”
Distribution Functions for Largest Eigenvalues and Their Applications (2002, background): “ Jockush, Propp and Shor [21] (see also [11]) that as n → ∞ the boundary between the temperate zone and the polar zones (appropriately scaled) converges to a circle (Arctic Circle Theorem). Johansson [24] proved that the ﬂuctuations about this limiting circle are F2. 2.4. Statistics Johnstone [27] c”
Large n Limit of Gaussian Random Matrices with External Source, Part I (2004, result): “the recent works [2], [ 20 ], [22], [28] and references cited therein.”