Results 1  10
of
97
Solving the KPZ equation
 Ann. of Math
, 2013
"... We introduce a new concept of solution to the KPZ equation which is shown to extend the classical ColeHopf solution. This notion provides a factorisation of the ColeHopf solution map into a “universal ” measurable map from the probability space into an explicitly described auxiliary metric space, ..."
Abstract

Cited by 57 (9 self)
 Add to MetaCart
(Show Context)
We introduce a new concept of solution to the KPZ equation which is shown to extend the classical ColeHopf solution. This notion provides a factorisation of the ColeHopf solution map into a “universal ” measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the ColeHopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class. As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a wellposedness result for the Fokker
Brownian gibbs property for airy line ensembles
"... 1 ≤ i ≤ N, conditioned not to intersect. The edgescaling limit of this system is obtained by taking a weak limit as N → ∞ of the collection of curves scaled so that the point (0, 2 1/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is ad ..."
Abstract

Cited by 30 (8 self)
 Add to MetaCart
(Show Context)
1 ≤ i ≤ N, conditioned not to intersect. The edgescaling limit of this system is obtained by taking a weak limit as N → ∞ of the collection of curves scaled so that the point (0, 2 1/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is added to each of the curves of this scaling limit, an xtranslation invariant process sometimes called the multiline Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which the curves are almost surely everywhere continuous and nonintersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with “wanderers ” and “outliers”. We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property – called the Brownian Gibbs property – of being invariant under the following action. Select an index 1 ≤ k ≤ N and erase Bk on a fixed time interval (a, b) ⊆ (−N, N); then replace this erased curve with a new curve on (a, b) according to the law of a Brownian bridge between the two existing endpoints ( a, Bk(a) ) and ( b, Bk(b) ) , conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edgescaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property. An immediate consequence of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the longstanding conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multiline Airy process, and readily yields several other interesting properties of this process. 1.
Free energy fluctuations for directed polymers in random media in 1+1 dimension
, 2012
"... ar ..."
(Show Context)
On integrability of zerorange chipping models with factorized steady state
, 2013
"... Conditions of integrability of general zero range chipping models with factorized steady state, which were proposed in [Evans, Majumdar, Zia 2004 J. Phys. A 37 L275], are examined. We find a threeparametric family of hopping probabilities for the models solvable by the Bethe ansatz, which include ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
(Show Context)
Conditions of integrability of general zero range chipping models with factorized steady state, which were proposed in [Evans, Majumdar, Zia 2004 J. Phys. A 37 L275], are examined. We find a threeparametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis we conjecture an integral formula for the Green function of evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.
Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
, 2014
"... We show that the geometric lifting of the RSK correspondence ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
We show that the geometric lifting of the RSK correspondence