Lattice Walks in a Weyl Chamber and Truncated Random Matrices

Let u(d, n) denote the number of permuations in the symmetric group Sn with no increasing subsequence of length greater than d. u(d, n) may alternatively be interpreted as the number of closed Z-lattice walks which begin and end at the origin and take n positive steps followed by n negative steps while remaining confined to the Weyl chamber W = {(t1, t2, . . . , td) ∈ R : t1 ≥ t2 ≥ · · · ≥ td}. It is well known that u(d, n) can be represented by a unitary matrix integral: u(d, n) = ∫ U(d) |Trace(V )|dV. Using this representation, we define a one-parameter family of deformations u(d, n, q) of u(d, n) depending on a real parameter q ≥ 0. At integer values of q, u(d, n, q) has the following combinatorial interpretation: it is the number of Z-lattice walks from the origin to the point (q, q, . . . , q) which take n negative steps, and remain in W. Regev gave an asymptotic formula for u(d, n) for d fixed and n→∞. We prove that u(d, n, q) can be expressed as an average over an ensemble of truncated random unitary matrices, and use this to give an asymptotic formula for u(d, n, q) with d, n fixed and q →∞ analagous to Regev’s. 0.1 Increasing Subsequences Let σ be a permutation in the symmetric group Sn. An increasing subsequence of σ is a collection of pairs (i1, σ(i1)), . . . , (il, σ(il)) satisfying 1 ≤ i1 < · · · < il ≤ n and 1 ≤ σ(i1) < · · · < σ(il) ≤ n. l is called the length of the increasing subsequence. The increasing subsequence problem is concerned with determining the distribution of the length of the longest increasing subsequence of a permutation chosen uniformly at random from Sn. A recent survey of the longest increasing subsequence problem which contains a comprehensive list of references is [26]. 0.2 Schensted’s correspondence and lattice paths Let Sd(n) denote the subset of Sn consisting of those permutations with no increasing subsequence of length greater than d, and let u(d, n) = |Sd(n)|. Schensted found a bijection between Sd(n) and the set of ordered pairs (P,Q) of standard Young tableaux on the same shape with entries 1, 2, . . . , n having at most d rows [23]. Consequently, u(d, n) = ∑