Limiting Distributions for a Polynuclear Growth Model with External Sources

By Jinho Baik, Eric M. Rains. 2000

285 citations recorded by the source corpus.

The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by Prähofer and Spohn. Depending on the strength of the sources, the limiting distribution functions are either the Tracy–Widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/28f9d5829aa74149f4d7a06865d8ecb10e9c5f76

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Citations in the corpus

Tropical Combinatorics and Whittaker functions (2011, background): “s as in equation (5.9). For (n,N) = tC~ and suitably chosen coeﬃcients c1 = c1(µ,ν) and c2 = c2(µ,ν): (5.13) lim t→∞ P nlogz N,1(n) −tc1 c2t1/3 ≤s o = F0(s) where F0(s) is the Baik-Rains distribution [9, 39]. One can also state similar conjectures for other boundary conditions which should limit to ”
Current Fluctuations for the Totally Asymmetric Simple Exclusion Process (2001, methodology): “In our context we follow [5] in the definition of the distribution functions, who rely on the established conventions for the Painlevé II Riemann-Hilbert problem.”
Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process (2005, methodology): “Baik and Rains [2], see also [18], use the Riemann-Hilbert techniques and arrive at a limit function, gBR, which is given as the solution of a set of differential equations(2).”
Limits of spiked random matrices I (2010, methodology): “They are proved in Baik and Rains [11] except for (iv) which goes back to Deift and Zhou [18].”
Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices (2005, background): “In the context of symmetrized random permutations and last passage percolation models, [4, 3] showed, among other things, the k = 1 case of Corollary 1.”
Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models (2000, background): “Further work by Johansson [29], Prähofer and Spohn [35, 36] and Baik and Rains [8] strongly suggests the universal nature of these RMT distribution functions.”
Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence (2012, background): “To add, many other situations have been considered in analytical work and shown to have a variety of intriguing distributions, such as the TW distribution for Gaussian symplectic ensemble [72, 82] and that for (GOE)2 [5, 72],12 expected under certain conditions with boundary or external sources.”
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices (2004, methodology): “The last passage percolation model considered in [5, 33] has some relevance to our problem; see Section 6 below.”
Fluctuation Properties of the TASEP with Periodic Initial Configuration (2006, background): “EN−1 0 0 −W[1,2) 0 · · · 0 0 0 0 −W[2,3) .”
Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function (2007, unknown): “6) of [3]), the function R(x) can also be written as R(x) = (q′(x))2 − x(q(x))(2) − (q(x))(4).”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, background): “However, using f (x) and g(x) associated with the Painlevé II equation (43), the Baik-Rains distribution can be expressed as [50, 69] Prob[χ0 ≤ x] ≡ F0(x) = [ 1 − (x + 2 f ′′(x) + 2g′′(x))g′(x)] e−2 f (x)−g(x).”
Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension (2012, methodology): “Information about these statistics originated from the analysis of a few solvable zero temperature, β = ∞, models [8, 10, 12, 17, 18, 38].”
Introduction to KPZ (2011, background): “The marginal Astat(0) has the Baik-Rains distribution F0 [9], [55].”
The Kardar-Parisi-Zhang equation and universality class (2011, unknown): “= −X X<−1 T1+X2 2 X∈[−1,0] T/2 X>0 h¯(T,TX) = −X X<−1 T1+X2 2 X∈[−1,0] T/2 X>0 h¯(T,TX) = T/2 Fluctuations •One pt: FGUE [9, 93] •Multi pt: Airy2 [137, 94, 29] •One pt: F0 [11, 69] •Multi pt: Airystat [10] •One pt: FGOE [12, 13, 70, 146] •Multi pt: Airy1 [31, 32] •One pt: (FGOE)2 [”
Extreme value statistics of correlated random variables: A pedagogical review (2019, background): “Quite remarkably, the same TW distributions (in particular for β = 1, 2) have emerged in a number of a priori unrelated problems [188] such as the longest increasing subsequence of random permutations [189], directed polymers [190, 191] and growth models [192–195] in the Kardar-Parisi-Zhang (KPZ) un”