Infinite wedge and random partitions

By A. Okounkov. 1999

Abstract. We use representation theory to obtain a number of exact results for random partitions. In particular, we prove a simple determinantal formula for correlation functions of what we call the Schur measure on partitions (which is a far reaching generalization of the Plancherel measure; see [3], [8]) and also observe that these correlations functions are $ \tau $-functions for the Toda lattice hierarchy. We also give a new proof of the formula due to Bloch and the author [5] for the so-called n-point functions of the uniform measure on partitions and comment on the local structure of a typical partition.

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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.”
Determinantal probability measures (2002, background): “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).”
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices (2004, background): “, [16, 21, 30]), some of which, by taking the above limit, would yield an alternative derivation of Proposition 2.”
Random matrices and determinantal processes (2005, methodology): “The probability measure on the points x 0 > � � � > x 0 (M = N) is the so-called Schur measure, [ 28 ].”
Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process (2005, background): “One could also arrive at an equivalent kernel by taking the exponential limit of the geometric case studied by Okounkov in [17].”
Macdonald processes (2011, methodology): “Setting q = t, both P and Q become Schur functions and the processes become the well-known Schur processes introduced in [82, 83].”
Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram (2001, methodology): “and we give an explicit contour integral representation for the correlation kernel K; see Theorem 1. This theorem is a generalization of Theorem 2 in [ 16 ].”
Fermionic representation for basic hypergeometric functions related to Schur polynomials (2000, methodology): “In [ 61 ] the interpretation of the Schur functions as a measure on partitions is explained, thus we have an additional interpretation of hypergeometric functions as generating functions for certain probabilities described in [61].”
Gromov-Witten theory, Hurwitz theory, and completed cycles (2002, unknown): “2) has been previously computed in [1], see also [27, 10].”
The importance of the Selberg integral (2007, unknown): “, [83, 128].”
Tropical Combinatorics and Whittaker functions (2011, background): “Such probability measures are known as Schur measures (see [49]).”
Seiberg-Witten theory and random partitions (2003, background): “In particular, the integrable structures of random matrices are preserved and, in fact, become more natural and transparent for the Plancherel measure, see [34][35].”