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Searching for the Tracy-Widom distribution in nonequilibrium processes  [PDF]
Christian B. Mendl,Herbert Spohn
Physics , 2015,
Abstract: While originally discovered in the context of the Gaussian Unitary Ensemble, the Tracy-Widom distribution also rules the height fluctuations of growth processes. This suggests that there might be other nonequilibrium processes in which the Tracy-Widom distribution plays an important role. In our contribution we study one-dimensional systems with domain wall initial conditions. For an appropriate choice of parameters the profile develops a rarefaction wave, while maintaining the initial equilibrium states far to the left and right, which thus serve as infinitely extended thermal reservoirs. For two distinct model systems we will demonstrate that the properly projected time-integrated current has a deterministic contribution, linear in time $t$, and fluctuations of size $t^{1/3}$ with a Tracy-Widom distributed random amplitude.
Painlevé representation of Tracy-Widom$_β$ distribution for $β = 6$  [PDF]
Igor Rumanov
Physics , 2014,
Abstract: In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of $\beta$. Using this general result, the case $\beta=6$ is further considered here. This is the smallest even $\beta$, when the corresponding Lax pair and its relation to Painlev\'e II (PII) have not been known before, unlike cases $\beta=2$ and $4$. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ODE for the logarithmic derivative of Tracy-Widom distribution for $\beta=6$ involving the PII function in the coefficients, is found, which allows one to compute asymptotics for the distribution function. The ODE is a consequence of a linear system of three ODEs for which the local Painlev\'e analysis yields series solutions with exponents in the set $4/3$, $1/3$ and $-2/3$.
The right tail exponent of the Tracy-Widom-beta distribution  [PDF]
Laure Dumaz,Bálint Virág
Mathematics , 2011, DOI: 10.1214/11-AIHP475
Abstract: The Tracy-Widom beta distribution is the large dimensional limit of the top eigenvalue of beta random matrix ensembles. We use the stochastic Airy operator representation to show that as a tends to infinity the tail of the Tracy Widom distribution satisfies P(TW_beta > a) = a^(-3/4 beta+o(1)) exp(-2/3 beta a^(3/2)).
A determinantal formula for the GOE Tracy-Widom distribution  [PDF]
Patrik L. Ferrari,Herbert Spohn
Mathematics , 2005, DOI: 10.1088/0305-4470/38/33/L02
Abstract: Investigating the long time asymptotics of the totally asymmetric simple exclusion process, Sasamoto obtains rather indirectly a formula for the GOE Tracy-Widom distribution. We establish that his novel formula indeed agrees with more standard expressions.
Numerical study of higher order analogues of the Tracy-Widom distribution  [PDF]
Tom Claeys,Sheehan Olver
Mathematics , 2011,
Abstract: We study a family of distributions that arise in critical unitary random matrix ensembles. They are expressed as Fredholm determinants and describe the limiting distribution of the largest eigenvalue when the dimension of the random matrices tends to infinity. The family contains the Tracy-Widom distribution and higher order analogues of it. We compute the distributions numerically by solving a Riemann-Hilbert problem numerically, plot the distributions, and discuss several properties that they appear to exhibit.
Higher order analogues of the Tracy-Widom distribution and the Painleve II hierarchy  [PDF]
T. Claeys,A. Its,I. Krasovsky
Mathematics , 2009,
Abstract: We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painlev\'e I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlev\'e II hierarchy. In addition we compute large gap asymptotics for the Fredholm determinants.
An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution  [PDF]
Satya N. Majumdar,Sergei Nechaev
Mathematics , 2003, DOI: 10.1103/PhysRevE.69.011103
Abstract: We compute exactly the asymptotic distribution of scaled height in a (1+1)--dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in $(1+1)$ dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all share the same universal Tracy-Widom distribution for the suitably scaled height variables.
Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution  [PDF]
Marco Chiani
Statistics , 2012, DOI: 10.1016/j.jmva.2014.04.002
Abstract: We derive efficient recursive formulas giving the exact distribution of the largest eigenvalue for finite dimensional real Wishart matrices and for the Gaussian Orthogonal Ensemble (GOE). In comparing the exact distribution with the limiting distribution of large random matrices, we also found that the Tracy-Widom law can be approximated by a properly scaled and shifted Gamma distribution, with great accuracy for the values of common interest in statistical applications.
Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers  [PDF]
Victor Dotsenko
Physics , 2010, DOI: 10.1209/0295-5075/90/20003
Abstract: The distribution function of the free energy fluctuations in one-dimensional directed polymers with $\delta$-correlated random potential is studied by mapping the replicated problem to a many body quantum boson system with attractive interactions. Performing the summation over the entire spectrum of excited states the problem is reduced to the Fredholm determinant with the Airy kernel which is known to yield the Tracy-Widom distribution
Role of the Tracy-Widom distribution in the finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass  [PDF]
Michele Castellana,Elia Zarinelli
Physics , 2011, DOI: 10.1103/PhysRevB.84.144417
Abstract: We investigate the finite-size fluctuations due to quenched disorder of the critical temperature of the Sherrington-Kirkpatrick spin glass. In order to accomplish this task, we perform a finite-size analysis of the spectrum of the susceptibility matrix obtained via the Plefka expansion. By exploiting results from random matrix theory, we obtain that the fluctuations of the critical temperature are described by the Tracy-Widom distribution with a non-trivial scaling exponent 2/3.
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