Discrete Polynuclear Growth and Determinantal Processes

By K. Johansson. 2002

465 citations recorded by the source corpus.

AbstractWe consider a discrete polynuclear growth (PNG) process and prove a functional limit theorem for its convergence to the Airy process. This generalizes previous results by Prähofer and Spohn. The result enables us to express the F1 GOE Tracy- Widom distribution in terms of the Airy process. We also show some results, and give a conjecture, about the transversal fluctuations in a point to line last passage percolation problem. Furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/01d5fdf8c727c64f2b39ad745a59b00ece589811

How researchers see this paper

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

In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions (2010, methodology): “In the case of TASEP, the one point distribution limit has been xtended to a process level limit.”
Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence (2012, background): “Theoretically, the GOE TW distribution was indeed identified analytically in the maximal height of the curved PNG interfaces at infinite time, or, equivalently, in the maximal position of the infinitely many non-intersecting Brownian particles with fixed end points (called the vicious walkers) [22, ”
Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension (2012, background): “For these models, one indeed finds ξ = 2/3, χ = 1/3, and although the existence of the full space-time limit under these scalings is not known, for fixed time τ > 0 the limiting spatial process is the Airy2 process [42, 53].”
The Kardar-Parisi-Zhang equation and universality class (2011, background): “where A2 represents the Airy2 process which is stationary, continuous [94], locally absolutely continuous to Brownian motion [48], has FGUE as its one point marginal and has finite dimensional distributions which can be written in terms of Fredholm determinants [137, 94] or certain PDEs [1, 167].”
Introduction to KPZ (2011, background): “Along with another solvable discretization called PNG (polynuclear growth model) [77], [76], [101] it has been the basis of the recent breakthroughs.”
Brownian Gibbs property for Airy line ensembles (2011, methodology): “We also obtain a proof of the long-standing conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point.”
Determinantal point processes (2018, result): “asily carried over to general state spaces X(j). Theorem 4.2 is often referred to as the Eynard-Mehta theorem, it was proved in [Eyn98] and also independently in [Nag98]. Other proofs can be found in [Joh03], [Tra04], [Bor05]. Thealgebraically “nice” case of theEynard-Mehta theorem, which e.g. takes”
The KPZ fixed point (2016, background): “msup N!1limsup "!0 P sup x2[ M;M] h "(x) N = 0. For the other direction, we bound h "(t;x) below by the narrow wedge solution h (t;x) centered at x ". By results of Johansson [Joh03], h"(1;x) + (x x ")2 converges uniformly on compact sets to the (rescaled) Airy 2 proc”
Stochastic six-vertex model (2014, background): “ , and vary 2R, then the (proper centered and scaled) random process H(Lx+L2=3;Ly;!) converges in distribution to the Airy 2 process (i.e. the top line of the Airy line ensemble). We refer to [PS2],[J2],[J3],[BF1] for similar statements for the interacting particle systems related to the determinant”
Random Skew Plane Partitions and the Pearcey Process (2005, background): “at a general point of the frozen boundary, suitably scaled, the correlation are given by the extended Airy kernel [17, 9, 19]; 3.”
Large time asymptotics of growth models on space-like paths I: PushASEP (2007, background): “23) in the sense of finite dimensional distributions, where A2 is the Airy2 process (see [18; 14; 11] and Section 2.”
Fluctuation Properties of the TASEP with Periodic Initial Configuration (2006, background): “Determinantal form of correlation functions appeared in different contexts [4,7,13,16,25].”