Random matrices and determinantal processes

By K. Johansson. 2005

315 citations recorded by the source corpus.

We survey recent results on determinantal processes, random growth, random tilings and their relation to random matrix theory.

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): “Aztec diamond tilings [Johansson, 2005a] The Aztec diamond is a diamond-shaped union of lattice squares, as depicted in Figure 5a. (Half of the squares have been colored gray in a checkerboard pattern.)”
Fluctuation Properties of the TASEP with Periodic Initial Configuration (2006, background): “For further references and details on determinantal point processes we refer the reader to surveys [14, 17, 22, 37] and the lecture notes [38].”
Stationary cocycles and Busemann functions for the corner growth model (2015, unknown): “ODEL 3 random matrix theory. When the weights have exponential or geometric distribution the model is exactly solvable, and it is possible to derive exact ﬂuctuation exponents and limit distributions [8, 31, 32]. In these cases the cocycles we construct have explicit product form distributions. The ”
Multiple orthogonal polynomial ensembles (2009, background): “[ 45 , 63], This means that there is a correlation kernel Kn(x, y) so that”
Determinantal point processes (2018, background): “articles of diﬀerent kinds attracted. Nowadays, the expression ‘determinantal point process (or ﬁeld)’ is standard. There are several excellent surveys of the subject available, see [Sos00], [Lyo03], [Joh05], [Ko¨n05], [Hou06], [Sos06]. Thereadermay ﬁndthere adetailed discussion of probabilistic pro”
Brownian Gibbs property for Airy line ensembles (2011, background): “Introduction One-dimensional Markov processes (such as random walks or Brownian motion) conditioned not to intersect form an important class of models which arise in the study of random matrix theory, growth processes, directed polymers, tilings and certain problems in combinatorics and representati”
Growth of balls of holomorphic sections and energy at equilibrium (2008, methodology): “(4.9) We will rely on the following formula for this L 2 -norm, which is well-known in the context of determinantal point processes (compare [Dei99] p.103, [Joh05] Propo-sition 2.10) and is also familiar in quantum mechanics (Slater determinants).”
Eigenvalues of GUE Minors (2006, background): “For that we refer to any or all of the following references: (DVJ88, Ch. 9.1, A2.1), ( Joh05c ; Sos00).”
Large time asymptotics of growth models on space-like paths I: PushASEP (2007, methodology): “Algebraically, we derive a determinantal formula for the distribution of the N -particle PushASEP with an arbitrary fixed initial condition, and we also represent this distribution as a gap probability for a (possibly, signed) determinantal point process (see [17; 12; 21; 16; 22] for information on ”
The Kardar-Parisi-Zhang equation and universality class (2011, background): “Though not directly addressed here, the study of these systems is closely related to and influenced by problems in random matrix theory, non-intersecting path ensembles, random tilings and certain combinatorial problems involving asymptotic representation theory [28, 71, 95, 158].”
On adding a list of numbers (and other one-dependent determinantal processes) (2009, background): “These include: the eigenvalues of various ensembles of random matrices (Dyson [25, 26, 27], Mehta [55]); the presence or absence of edges if a random spanning tree is chosen in a graph (Burton‐Pemantle [10], Lyons [48]); random tilings and growth models (Johansson [ 44 ]); the structure of random pa”
The One-Dimensional KPZ Equation and Its Universality Class (2015, background): “rticles on a one-dimensional lattice Zperforming nearest neighbor random walks, with an asymmetric jump rate and subject to the exclusion rule of at most one particle per site. The jump rates are p ∈ [0,1] for jumps to the right, i.e. from j to j +1, and q = 1−p for jumps to the left, all jump trial”