Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation

We study the combinatorial structure of the irreducible characters of the classical groups ${\rm GL}_{n}(\mathbb{C})$, ${\rm SO}_{2n+1}(\mathbb{C})$, ${\rm Sp}_{2n}(\mathbb{C})$, ${\rm SO}_{2n}(\mathbb{C})$ and the "non-classical" odd symplectic group ${\rm Sp}_{2n+1}(\mathbb{C})$, finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$ and between characters of ${\rm SO}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$. We identify the first family as a one-parameter specialization of Koornwinder polynomials, for which we thus provide a novel combinatorial structure; on the other hand, the second family appears to be new. We next develop a method of Gelfand-Tsetlin pattern decomposition to establish identities between all these polynomials that, in the case of characters, can be viewed as describing the decomposition of irreducible representations of the groups when restricted to certain subgroups. Through these formulas we connect orthogonal and symplectic characters, and more generally the interpolating polynomials, to LPP models with various symmetries, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains [BR01a]. Taking the scaling limit of the LPP models, we finally provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians.Comment: 60 pages, 11 figures. Typos corrected and a few remarks adde

GOE and Airy2→1{\rm Airy}_{2\to 1} marginal distribution via symplectic Schur functions

We derive Sasamoto's Fredholm determinant formula for the Tracy-Widom GOE distribution, as well as the one-point marginal distribution of the ${\rm Airy}_{2\to1}$ process, originally derived by Borodin-Ferrari-Sasamoto, as scaling limits of point-to-line and point-to-half-line last passage percolation with exponentially distributed waiting times. The asymptotic analysis goes through new expressions for the last passage times in terms of integrals of (the continuous analog of) symplectic and classical Schur functions, obtained recently in [BZ19a].Comment: 19 pages, 2 figures. Typos corrected, references adde

Variants of geometric RSK, geometric PNG and the multipoint distribution of the log-gamma polymer

We show that the reformulation of the geometric Robinson-Schensted-Knuth (gRSK) correspondence via local moves, introduced in \cite{OSZ14} can be extended to cases where the input matrix is replaced by more general polygonal, Young-diagram-like, arrays of the form \polygon. We also show that a rearrangement of the sequence of the local moves gives rise to a geometric version of the polynuclear growth model (PNG). These reformulations are used to obtain integral formulae for the Laplace transform of the joint distribution of the point-to-point partition functions of the log-gamma polymer at different space-time points. In the case of two points at equal time $N$ and space at distance of order $N^{2/3}$, we show formally that the joint law of the partition functions, scaled by $N^{1/3}$, converges to the two-point function of the Airy processComment: 44 pages. Proposition 3.4 and Theorem 3.5 are now stated in a more general form and some more minor changes are made (most of them following suggestions by a referee). To appear at IMR

Subgaussian concentration and rates of convergence in directed polymers

We consider directed random polymers in (d+1) dimensions with nearly gamma i.i.d. disorder. We study the partition function ZN,ω and establish exponential concentration of log ZN,ω about its mean on the subgaussian scale √N/log N . This is used to show that E[log ZN,ω] differs from N times the free energy by an amount which is also subgaussian (i.e. o(√N)), specifically O(√N/logN log log N)

Universality in marginally relevant disordered systems

We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2+1), the long-range directed polymer model with Cauchy tails in dimension (1+1) and the disordered pinning model with tail exponent 1/2. We show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit: a log-normal random field with a multi-scale correlation structure, which undergoes a phase transition as the disorder strength varies. As a by-product, we show that the solution of the two-dimensional Stochastic Heat Equation, suitably regularized, converges to the same limit. The proof, which uses the celebrated Fourth Moment Theorem, reveals an interesting chaos structure shared by all models in the above class.Comment: 49 pages. Fixed typos, added comments and references. To appear in Ann. Appl. Proba

Geometric RSK correspondence, Whittaker functions and symmetrized random polymers

We show that the geometric lifting of the RSK correspondence introduced by A.N. Kirillov (2001) is volume preserving with respect to a natural product measure on its domain, and that the integrand in Givental's integral formula for GL(n,R)-Whittaker functions arises naturally in this context. Apart from providing further evidence that Whittaker functions are the natural analogue of Schur polynomials in this setting, our results also provide a new `combinatorial' framework for the study of random polymers. When the input matrix consists of random inverse gamma distributed weights, the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of Whittaker functions. Next we restrict the geometric RSK mapping to symmetric matrices and show that the volume preserving property continues to hold. We determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal, with an additional factor on the main diagonal. The third combinatorial mapping studied is a variant of the geometric RSK mapping for triangular arrays, which is again showed to be volume preserving. This leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal. We also show that the analogues of the Cauchy-Littlewood identity in the setting of this paper are equivalent to a collection of Whittaker integral identities conjectured by Bump (1989) and Bump and Friedberg (1990) and proved by Stade (2001, 2002). Our approach leads to new `combinatorial' proofs and generalizations of these identities, with some restrictions on the parameters.Comment: v2: significantly extended versio

Some algebraic structures in the KPZ universality

We review some algebraic and combinatorial structures that underlie models in the KPZ universality class. Emphasis is given on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov and we present how these are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct stochastic dynamics on two dimensional arrays called Gelfand-Tsetlin patterns, in ways that couple different one dimensional stochastic processes. The goal of the notes is to expose some of the overarching principles, that have driven a significant number of developments in the field, as a unifying theme.Comment: 75 pages, several figures. This is a review / lecture notes material. Some references adde