A Fredholm Determinant Representation in ASEP

By Craig A. Tracy, Harold Widom. 2008

138 citations recorded by the source corpus.

In previous work (Tracy and Widom in Commun. Math. Phys. 279:815–844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice ℤ. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers ℤ+ and consider the distribution function for the mth particle from the left. In Tracy and Widom (Commun. Math. Phys. 279:815–844, 2008) an infinite series of multiple integrals was derived for the distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. We use this determinant representation to derive (non-rigorously, at this writing) a scaling limit.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3c7042f4da0128b57eba2f085456a1995a6880a6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Height Fluctuations for the Stationary KPZ Equation (2014, methodology): “This provides an alternative to the methods of [79–81] as well as a rigorous regularization of the replica method used in [40] and [32].”
The Kardar-Parisi-Zhang equation and universality class (2011, methodology): “Let us briefly remark on the approach Tracy and Widom used in deriving this formula in [163, 164, 165].”
Introduction to KPZ (2011, background): “In a series of articles starting with [114, 115] C.”
Directed polymers and the quantum Toda lattice (2009, background): “It will be interesting to relate this, in a suitable scaling limit, to the “crossover distributions” recently introduced in the context of KPZ and the stochastic heat equation by Sasamoto and Spohn [49–52] and Amir, Corwin and Quastel [1], building on recent work of Tracy and Widom [58–61] on the as”
From duality to determinants for q-TASEP and ASEP (2012, methodology): “The ASEP Fredholm determinant Tracy and Widom derived in [41, 42] is also of this type (and in fact, after inverting the eτ -Laplace transform we recover the same formula as in [41, 42]).”
Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions (2010, background): “This formula was stated in [32] in the form below, and was developed in the three papers [30, 31, 32].”
On a family of symmetric rational functions (2014, background): “A few years after the work of Tracy and Widom, the case of α = β = γ = 1, δ = 0, was considered by Borodin-Corwin-Petrov-Sasamoto [5] in connection with the so-called q-TASEP and a q-Boson particle system (here ‘T’ in ‘TASEP’ stands for ‘totally’, and total asymmetry means that particles in the syst”
Macdonald processes (2011, methodology): “Those works took asymptotics of Tracy and Widom’s ASEP formulas [96, 97, 98] and rely on the fact that weakly scaled ASEP converges to the KPZ equation [13, 4].”
Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension (2012, methodology): “For m = 0 the free energy probability distribution was discovered simultaneously and independently in both [5, 56] and rigorously proved in [5] via Tracy and Widom’s ASEP formulas [63–65, 67]; the above Laplace transform formula was soon after (non-rigorously) derived from the replica trick approach”