280 research outputs found
Geometric RSK and the Toda lattice
We relate a continuous-time version of the geometric RSK correspondence to
the Toda lattice, in a way which can be viewed as a semi-classical limit of a
recent result by the author which relates the continuous-time geometric RSK
mapping, with Brownian motion as input, to the quantum Toda lattice.Comment: v2: minor correction
Random matrices, non-colliding processes and queues
This is survey of some recent results connecting random matrices,
non-colliding processes and queues.Comment: To appear in Seminaire de Probabilites XXXV
A q-weighted version of the Robinson-Schensted algorithm
We introduce a q-weighted version of the Robinson-Schensted (column
insertion) algorithm which is closely connected to q-Whittaker functions (or
Macdonald polynomials with t=0) and reduces to the usual Robinson-Schensted
algorithm when q=0. The q-insertion algorithm is `randomised', or `quantum', in
the sense that when inserting a positive integer into a tableau, the output is
a distribution of weights on a particular set of tableaux which includes the
output which would have been obtained via the usual column insertion algorithm.
There is also a notion of recording tableau in this setting. We show that the
distribution of weights of the pair of tableaux obtained when one applies the
q-insertion algorithm to a random word or permutation takes a particularly
simple form and is closely related to q-Whittaker functions. In the case , the q-insertion algorithm applied to a random word also provides a new
framework for solving the q-TASEP interacting particle system introduced (in
the language of q-bosons) by Sasamoto and Wadati (1998) and yields formulas
which are equivalent to some of those recently obtained by Borodin and Corwin
(2011) via a stochastic evolution on discrete Gelfand-Tsetlin patterns (or
semistandard tableaux) which is coupled to the q-TASEP process. We show that
the sequence of P-tableaux obtained when one applies the q-insertion algorithm
to a random word defines another, quite different, evolution on semistandard
tableaux which is also coupled to the q-TASEP process
On the free energy of a directed polymer in a Brownian environment
We prove a formula conjectured in O'Connell and Yor (2001) for the free
energy density of a directed polymer in a Brownian environment in 1+1
dimensions.Comment: 15 pages. To appear in Markov Processes and Related Filelds (J.T.
Lewis special edition
Littelmann paths and brownian paths
We study some path transformations related to Littelmann path model and their
applications to representation theory and Brownian motion in a Weyl chamber.Comment: 30 pages, 1 figur
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
Free fermions and the classical compact groups
There is a close connection between the ground state of non-interacting
fermions in a box with classical (absorbing, reflecting, and periodic) boundary
conditions and the eigenvalue statistics of the classical compact groups. The
associated determinantal point processes can be extended in two natural
directions: i) we consider the full family of admissible quantum boundary
conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded
interval, and the corresponding projection correlation kernels; ii) we
construct the grand canonical extensions at finite temperature of the
projection kernels, interpolating from Poisson to random matrix eigenvalue
statistics. The scaling limits in the bulk and at the edges are studied in a
unified framework, and the question of universality is addressed. Whether the
finite temperature determinantal processes correspond to the eigenvalue
statistics of some matrix models is, a priori, not obvious. We complete the
picture by constructing a finite temperature extension of the Haar measure on
the classical compact groups. The eigenvalue statistics of the resulting grand
canonical matrix models (of random size) corresponds exactly to the grand
canonical measure of non-interacting free fermions with classical boundary
conditions.Comment: 35 pages, 5 figures. Final versio
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