Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram

By A. Okounkov, N. Reshetikhin. 2001

435 citations recorded by the source corpus.

Schur process is a time-dependent analog of the Schur measure on partitions studied in math.RT/9907127. Our first result is that the correlation functions of the Schur process are determinants with a kernel that has a nice contour integral representation in terms of the parameters of the process. This general result is then applied to a particular specialization of the Schur process, namely to random 3-dimensional Young diagrams. The local geometry of a large random 3-dimensional diagram is described in terms of a determinantal point process on a 2-dimensional lattice with the incomplete beta function kernel (which generalizes the discrete sine kernel). A brief discussion of the universality of this answer concludes the paper.

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

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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.”
Universality for mathematical and physical systems (2006, positive): “ the KPZ class ([PraSpo], [FerPra]), and between random matrix theory and equilibrium crystals with short range interactions ([CerKen], [FerSpo], UNIVERSALITY FOR MATHEMATICAL AND PHYSICAL SYSTEMS 21 [OkoRes], [FerPraSpo]). In addition, for applications to principal component analysis in statistics ”
Determinantal point processes (2018, background): “x+1Jy−1 x−y = X k∈Z′ >0 Jx+kJy+k, 13 where Jk = Jk(2θ) are the J-Bessel functions. This is the so-called discrete Bessel kernel that was ﬁrst obtained in [Bor00b], [Joh01]. We refer to [Oko01] and [Oko03] for far-reaching generalizations of Theorem 6.1, and to [Lyt02] for a general construction o”
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Scale Invariance of the PNG Droplet and the Airy Process (2001, background): “e integral dominates the factor e−λ′y. This implies the bound |R(λ′,λ)| ≤ ce−γ( |λ′ + ) (B.2) with suitable constants c,γ>0. Hence Ris of trace class. Note added in proof: In their recent preprint [25] Okounkov and Reshetikhin consider the (1,1,1) interface of the three-dimensional Ising model at”
Fluctuation Properties of the TASEP with Periodic Initial Configuration (2006, background): “Determinantal form of correlation functions appeared in different contexts [4,7,13,16,25].”