Symmetric functions and P-recursiveness

Symmetrized Models of Last Passage Percolation and Non-Intersecting Lattice Paths

“…Define the length of the extended path as the number of points it contains, and denote by l = l (λ) the stochastic variable specifying the maximum of the lengths of all possible extended paths (see Figure 1). Then it is known from the work of Gessel [11] and Rains [19] (see also [5]) that Pr(l ≤ l) = l j=1 e √ λ cos θj U(l) (1.1) where the average is with respect to the eigenvalue probability density function (p.d.f.) of random matrices chosen uniformly at random from the group U (l).…”

Vicious Walkers and Random Contraction Matrices

“…This is precisely Gessel's identity, the original proof of which appears in [18]. Alternative proof of this result were later given by Gessel, Weinstein, and Wilf [41], Tracy and Widom [39], and Xin [42].…”

“…The identity between the matrix integral and the generating series of u d (n) is due to Rains [33], while the identity between the series and the determinant is due to Gessel [18]. Both were discovered in the context of the enumeration of permutations with bounded increasing subsequence length.…”

Untitled

“…This identity can be seen as a special case of a result of Gessel [17]. Incidently, the density exp(α cos(·)) on the circle is the classical von Mises -Fisher or circular Normal distribution ( [23], Chapter 33).…”