3,271 research outputs found
From duality to determinants for q-TASEP and ASEP
We prove duality relations for two interacting particle systems: the
-deformed totally asymmetric simple exclusion process (-TASEP) and the
asymmetric simple exclusion process (ASEP). Expectations of the duality
functionals correspond to certain joint moments of particle locations or
integrated currents, respectively. Duality implies that they solve systems of
ODEs. These systems are integrable and for particular step and half-stationary
initial data we use a nested contour integral ansatz to provide explicit
formulas for the systems' solutions, and hence also the moments. We form
Laplace transform-like generating functions of these moments and via residue
calculus we compute two different types of Fredholm determinant formulas for
such generating functions. For ASEP, the first type of formula is new and
readily lends itself to asymptotic analysis (as necessary to reprove GUE
Tracy--Widom distribution fluctuations for ASEP), while the second type of
formula is recognizable as closely related to Tracy and Widom's ASEP formula
[Comm. Math. Phys. 279 (2008) 815--844, J. Stat. Phys. 132 (2008) 291--300,
Comm. Math. Phys. 290 (2009) 129--154, J. Stat. Phys. 140 (2010) 619--634]. For
-TASEP, both formulas coincide with those computed via Borodin and Corwin's
Macdonald processes [Probab. Theory Related Fields (2014) 158 225--400]. Both
-TASEP and ASEP have limit transitions to the free energy of the continuum
directed polymer, the logarithm of the solution of the stochastic heat equation
or the Hopf--Cole solution to the Kardar--Parisi--Zhang equation. Thus,
-TASEP and ASEP are integrable discretizations of these continuum objects;
the systems of ODEs associated to their dualities are deformed discrete quantum
delta Bose gases; and the procedure through which we pass from expectations of
their duality functionals to characterizing generating functions is a rigorous
version of the replica trick in physics.Comment: Published in at http://dx.doi.org/10.1214/13-AOP868 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Free energy fluctuations for directed polymers in random media in 1+1 dimension
We consider two models for directed polymers in space-time independent random
media (the O'Connell-Yor semi-discrete directed polymer and the continuum
directed random polymer) at positive temperature and prove their KPZ
universality via asymptotic analysis of exact Fredholm determinant formulas for
the Laplace transform of their partition functions. In particular, we show that
for large time tau, the probability distributions for the free energy
fluctuations, when rescaled by tau^{1/3}, converges to the GUE Tracy-Widom
distribution.
We also consider the effect of boundary perturbations to the quenched random
media on the limiting free energy statistics. For the semi-discrete directed
polymer, when the drifts of a finite number of the Brownian motions forming the
quenched random media are critically tuned, the statistics are instead governed
by the limiting Baik-Ben Arous-Peche distributions from spiked random matrix
theory. For the continuum polymer, the boundary perturbations correspond to
choosing the initial data for the stochastic heat equation from a particular
class, and likewise for its logarithm -- the Kardar-Parisi-Zhang equation. The
Laplace transform formula we prove can be inverted to give the one-point
probability distribution of the solution to these stochastic PDEs for the class
of initial data.Comment: 73 pages, 10 figure
Diffuse-Charge Dynamics in Electrochemical Systems
The response of a model micro-electrochemical system to a time-dependent
applied voltage is analyzed. The article begins with a fresh historical review
including electrochemistry, colloidal science, and microfluidics. The model
problem consists of a symmetric binary electrolyte between parallel-plate,
blocking electrodes which suddenly apply a voltage. Compact Stern layers on the
electrodes are also taken into account. The Nernst-Planck-Poisson equations are
first linearized and solved by Laplace transforms for small voltages, and
numerical solutions are obtained for large voltages. The ``weakly nonlinear''
limit of thin double layers is then analyzed by matched asymptotic expansions
in the small parameter , where is the
screening length and the electrode separation. At leading order, the system
initially behaves like an RC circuit with a response time of
(not ), where is the ionic diffusivity, but nonlinearity
violates this common picture and introduce multiple time scales. The charging
process slows down, and neutral-salt adsorption by the diffuse part of the
double layer couples to bulk diffusion at the time scale, . In the
``strongly nonlinear'' regime (controlled by a dimensionless parameter
resembling the Dukhin number), this effect produces bulk concentration
gradients, and, at very large voltages, transient space charge. The article
concludes with an overview of more general situations involving surface
conduction, multi-component electrolytes, and Faradaic processes.Comment: 10 figs, 26 pages (double-column), 141 reference
Biorthogonal polynomials associated with reflection groups and a formula of Macdonald
Dunkl operators are differential-difference operators on \b R^N which
generalize partial derivatives. They lead to generalizations of Laplace
operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on.
In this paper we introduce two systems of biorthogonal polynomials with respect
to Dunkl's Gaussian distributions in a quite canonical way. These systems,
called Appell systems, admit many properties known from classical Hermite
polynomials, and turn out to be useful for the analysis of Dunkl's Gaussian
distributions. In particular, these polynomials lead to a new proof of a
generalized formula of Macdonald due to Dunkl.
The ideas for this paper are taken from recent works on non-Gaussian white
noise analysis and from the umbral calculus.Comment: 14 pages, Latex2
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 and space at
distance of order , we show formally that the joint law of the
partition functions, scaled by , 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
- …