Painlevé Representation of Tracy–Widom $${_\beta}$$ β Distribution for $${\beta}$$ β = 6

Rumanov, Igor

doi:10.1007/s00220-015-2487-5

Cited by 34 publications

(33 citation statements)

References 55 publications

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“…Its cumulative distribution, F β = Prob[χ β ≤ x], is known as the β-Tracy-Widom (TW) distribution. For β = 1, 2, 4, F β can be written explicitly in terms of a special solution of a Painlevé equation [8,9] and for β = 6 in terms of an additional Painlevé transcendent [179,180]. For other values of β, it can be shown that F β describes the fluctuations of the ground state of a one-dimensional random Schrödinger operator, called the "stochastic Airy operator" [181,182].…”

Section: Extreme Statistics In Random Matrix Theorymentioning

confidence: 99%

Extreme value statistics of correlated random variables: A pedagogical review

2020

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Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of 'uncorrelated' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this review, we will first recall the classical EVS for uncorrelated variables and discuss the three universality classes of extreme value limiting distribution, known as the Gumbel, Fréchet and Weibull distribution. We then show that, for weakly correlated random variables with a finite correlation length/time, the limiting extreme value distribution can still be inferred from that of the uncorrelated variables using a renormalisation group-like argument. Finally, we consider the most interesting examples of strongly correlated variables for which there are very few exact results for the EVS. We discuss few examples of such strongly correlated systems (such as the Brownian motion and the eigenvalues of a random matrix) where some analytical progress can be made. We also discuss other observables related to extremes, such as the density of near-extreme events, time at which an extreme value occurs, order and record statistics, etc.

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Section: Extreme Statistics In Random Matrix Theorymentioning

confidence: 99%

Extreme value statistics of correlated random variables: A pedagogical review

2020

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“…During the last decade, there were written quite a lot of works on relations of the equations of IDM for Painlevé type ODEs with evolution linear equations of quantum mechanics and, starting from work [36], of quantum field theory [1]- [3], [7], [8], [14]- [17], [19]- [24], [26], [27], [32], [35]- [38], [41], [43]- [45].…”

Section: Introductionmentioning

confidence: 99%

“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

Pavlenko¹,

Suleǐmanov²

2017

Ufimskii Matematicheskii Zhurnal

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Abstract. We consider two compatible linear evolution equations with times 1 and 2 depending on two spatial variables. These evolution equations are the analogues of the non-stationary Schrödinger equations determined by the two Hamiltonians 7 2 +1 ( 1 , 2 , 1 , 2 , 1 , 2 ) ( = 1, 2) of the Hamilton system +1 . They arise by the formal replacement of the Planck constant by the imaginary unit. We construct explicit solutions of these analogues of Schrödinger equations in terms of the solutions of the corresponding linear systems of ordinary differential equations in the isomonodromic deformations method, whose compatibility condition is the Hamiltonian system 7 2 +1 . The key role in the construction of these explicit solutions is played by the change, which was used earlier in constructing the solutions of non-stationary Schrödinger equation determined by the Hamiltonians of isomonodromic Hamiltonian Garnier system with two degrees of freedom as well as of two isomonodromic degenerations of the latter. We discuss the applicability of this change for constructing the solutions to analogues of non-stationary Schrödinger equations determined by the Hamiltonians of the entire hierarchy of isomonodromic Hamiltonian systems with two degrees of freedom being the degenerations of this Garnier system. We mention also a relation of solutions to Hamilton systems +1 with some problems of modern nonlinear mathematical physics. In particular, we show that the solutions of these Hamiltonian systems are determined explicitly by the simultaneous solutions to the Korteweg-de Vries equation + + = 0 and a non-autonomous fifth order ordinary differential equations, which are used in universal description of the influence of a small dispersion on the transformation of weak hydrodynamical discontinuities into the strong ones.

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“…That same paper provides new proofs of the β = 2, 4 spiked soft edge laws using the analog of the PDE (1.12) derived in [3]. It is further worth mentioning that more recently Rumanov [23] has found a Painlevé II formula at the β = 6 soft edge (the first known outside β = 1, 2, 4) once more using the spiked/PDE picture of [3].…”

Section: Riccati Correspondence and Pdesmentioning

confidence: 77%

“…In brief, the introduction of the variable boundary condition produces the previously missing space-like variable in the corresponding PDE(s). These PDEs have subsequently been used to confirm a conjectured Painlevé formula for the spiked β = 4 TracyWidom law form [31], and more recently has led to Painlevé formulas at β = 6 [23] (the first appearance of such outside of β = 1, 2, 4).…”

Section: Riccati Correspondence and Pdesmentioning

confidence: 94%

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Spiking the random matrix hard edge

Ramı́rez

Rider

2016

Probab. Theory Relat. Fields

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We characterize the limiting smallest eigenvalue distributions (or hard edge laws) for sample covariance type matrices drawn from a spiked population. In the case of a single spike, the results are valid in the context of the general β ensembles. For multiple spikes, the necessary construction restricts matters to real, complex or quaternion (β = 1, 2, or 4) ensembles. The limit laws are described in terms of a random integral operators, and partial differential equations satisfied by the corresponding distribution functions are derived as corollaries. We also show that, under a natural limit, all spiked hard edge laws derived here degenerate to the critically spiked soft edge laws (or deformed Tracy-Widom laws). The latter were first described at β = 2 by Baik, Ben Arous, and Peché [2], and from a unified β random operator point of view by Bloemendal and Virág [3,4].

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Painlevé Representation of Tracy–Widom $${_\beta}$$ β Distribution for $${\beta}$$ β = 6

Cited by 34 publications

References 55 publications

Extreme value statistics of correlated random variables: A pedagogical review

Extreme value statistics of correlated random variables: A pedagogical review

“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

Spiking the random matrix hard edge

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Painlevé Representation of Tracy–Widom $${_\beta}$$ β Distribution for $${\beta}$$ β = 6

Cited by 34 publications

References 55 publications

Extreme value statistics of correlated random variables: A pedagogical review

Extreme value statistics of correlated random variables: A pedagogical review

&#x201C;Quantizations&#x201D; of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

Spiking the random matrix hard edge

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“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$