2,725 research outputs found
Nearest neighbor Markov dynamics on Macdonald processes
Macdonald processes are certain probability measures on two-dimensional
arrays of interlacing particles introduced by Borodin and Corwin
(arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative
specializations of the Macdonald symmetric functions and depend on two
parameters (q,t), where 0<= q, t < 1. Our main result is a classification of
continuous time, nearest neighbor Markov dynamics on the space of interlacing
arrays that act nicely on Macdonald processes.
The classification unites known examples of such dynamics and also yields
many new ones. When t = 0, one dynamics leads to a new integrable interacting
particle system on the one-dimensional lattice, which is a q-deformation of the
PushTASEP (= long-range TASEP). When q = t, the Macdonald processes become the
Schur processes of Okounkov and Reshetikhin (arXiv:math/0107056 [math.CO]). In
this degeneration, we discover new Robinson--Schensted-type correspondences
between words and pairs of Young tableaux that govern some of our dynamics.Comment: 90 pages; 13 figure
Inhomogeneous exponential jump model
We introduce and study the inhomogeneous exponential jump model - an
integrable stochastic interacting particle system on the continuous half line
evolving in continuous time. An important feature of the system is the presence
of arbitrary spatial inhomogeneity on the half line which does not break the
integrability. We completely characterize the macroscopic limit shape and
asymptotic fluctuations of the height function (= integrated current) in the
model. In particular, we explain how the presence of inhomogeneity may lead to
macroscopic phase transitions in the limit shape such as shocks or traffic
jams. Away from these singularities the asymptotic fluctuations of the height
function around its macroscopic limit shape are governed by the GUE Tracy-Widom
distribution. A surprising result is that while the limit shape is
discontinuous at a traffic jam caused by a macroscopic slowdown in the
inhomogeneity, fluctuations on both sides of such a traffic jam still have the
GUE Tracy-Widom distribution (but with different non-universal normalizations).
The integrability of the model comes from the fact that it is a degeneration
of the inhomogeneous stochastic higher spin six vertex models studied earlier
in arXiv:1601.05770 [math.PR]. Our results on fluctuations are obtained via an
asymptotic analysis of Fredholm determinantal formulas arising from contour
integral expressions for the q-moments in the stochastic higher spin six vertex
model. We also discuss "product-form" translation invariant stationary
distributions of the exponential jump model which lead to an alternative
hydrodynamic-type heuristic derivation of the macroscopic limit shape.Comment: 52 pages, 12 figure
Static quark anti-quark pair in SU(2) gauge theory
We study singlet and triplet correlation functions of static quark anti-quark
pair defined through gauge invariant time-like Wilson loops and Polyakov loop
correlators in finite temperature SU(2) gauge theory. We use the Luescher-Weisz
multilevel algorithm, which allows to calculate these correlators at very low
temperatures. We observe that the naive separation of singlet and triplet
states in general does not hold non-perturbatively, however, is recovered in
the limit of small separation and the temperature dependence of the
corresponding correlators is indeed very different.Comment: ReVTeX, 11 pages, 5 figure
Spectral theory for the q-Boson particle system
We develop spectral theory for the generator of the q-Boson (stochastic)
particle system. Our central result is a Plancherel type isomorphism theorem
for this system. This theorem has various implications. It proves the
completeness of the Bethe ansatz for the q-Boson generator and consequently
enables us to solve the Kolmogorov forward and backward equations for general
initial data. Owing to a Markov duality with q-TASEP, this leads to moment
formulas which characterize the fixed time distribution of q-TASEP started from
general initial conditions. The theorem also implies the biorthogonality of the
left and right eigenfunctions.
We consider limits of our q-Boson results to a discrete delta Bose gas
considered previously by van Diejen, as well as to another discrete delta Bose
gas that describes the evolution of moments of the semi-discrete stochastic
heat equation (or equivalently, the O'Connell-Yor semi-discrete directed
polymer partition function). A further limit takes us to the delta Bose gas
which arises in studying moments of the stochastic heat equation /
Kardar-Parisi-Zhang equation.Comment: 63 pages, 5 figure
Integrable probability: From representation theory to Macdonald processes
These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the (q, t)-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer's partition function.National Science Foundation (U.S.) (Grant DMS-1056390
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