Selberg integrals, Askey-Wilson polynomials and lozenge tilings of a hexagon with a triangular hole

We obtain an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. The complexity of our expression depends on the distance from the hole to the center of the hexagon. This proves and generalizes conjectures of Ciucu et al., who considered the case of plain enumeration when the triangle is located at or very near the center. Our proof uses Askey-Wilson polynomials as a tool to relate discrete and continuous Selberg-type integrals.Comment: 29 pages; minor changes from v

Elliptic rook and file numbers

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and later was extended to the q-case by Garsia and Remmel. We obtain similar results for our elliptic analogues of Garsia and Remmel's q-file numbers for skyline boards. We also provide an elliptic extension of the j-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings has been taken out to keep the length of this paper reasonable

Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices

We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as sums of squares).Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Schramm's formula for multiple loop-erased random walks

We revisit the computation of the discrete version of Schramm's formula for the loop-erased random walk derived by Kenyon. The explicit formula in terms of the Green function relies on the use of a complex connection on a graph, for which a line bundle Laplacian is defined. We give explicit results in the scaling limit for the upper half-plane, the cylinder and the Moebius strip. Schramm's formula is then extended to multiple loop-erased random walks.Comment: 59 pages, 19 figures. v2: reformulation of Section 2.3, minor correction

An elliptic extension of the multinomial theorem

We present a multinomial theorem for elliptic commuting variables. This result extends the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a simple elliptic star-triangle relation, ensuring the uniqueness of the normal form coefficients, and, for the recursion of the closed form elliptic multinomial coefficients, the Weierstra{\ss} type $\mathsf A$ elliptic partial fraction decomposition. From our elliptic multinomial theorem we obtain, by convolution, an identity that is equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel--Turaev ${}_{10}V_9$ summation, which in the trigonometric or basic limiting case reduces to Milne's type $\mathsf A$ extension of the Jackson ${}_8\phi_7$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, our derivation of the $\mathsf A_r$ Frenkel--Turaev summation constitutes the first combinatorial proof of that fundamental identity, and, at the same time, of important special cases including the $\mathsf A_r$ Jackson summation.Comment: 14 p

q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon's product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three two-dimensional families. For distributions from these families we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large, and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.Comment: 58 pages, v2: minor change