scholarly journals Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

2011 ◽  
Vol 309 (3) ◽  
pp. 793-833 ◽  
Author(s):  
M. Bertola ◽  
M. Cafasso
Keyword(s):  
Fredholm Determinants ◽  
Painlevé Ii

scholarly journals A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants

2021 ◽  
Author(s):  
Sofia Tarricone ◽  
Keyword(s):  
Hankel Operators ◽  
Lax Pair ◽  
Airy Functions ◽  
Fredholm Determinants ◽  
The Matrix ◽  
Painlevé Ii

We consider Fredholm determinants of matrix Hankel operators associated to matrix versions of the n-th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlevé II hierarchy, defined through a matrix-valued version of the Lenard operators. In particular, the Riemann-Hilbert techniques used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitly written in terms of the matrix-valued Lenard operators and some solutions of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy Hankel operators.

scholarly journals Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes

2019 ◽  
Author(s):  
Mattia Cafasso ◽  
Tom Claeys ◽  
Manuela Girotti
Keyword(s):  
Statistical Physics ◽  
Point Processes ◽  
Fredholm Determinant ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Contour Integrals ◽  
Painlevé Ii ◽  
Derivatives Of ◽  
Gap Probabilities ◽  
Logarithmic Derivatives

Abstract We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr [20]. In addition, we obtain asymptotics at $\pm \infty $ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

On Ermakov–Painlevé II systems. Integrable reduction

Meccanica ◽  
2016 ◽  
Vol 51 (12) ◽  
pp. 2967-2974 ◽  
Author(s):  
Colin Rogers ◽  
Wolfgang K. Schief
Keyword(s):  
Painlevé Ii

scholarly journals Bilinear discrete Painleve-II and its particular solutions

1995 ◽  
Vol 28 (12) ◽  
pp. 3541-3548 ◽  
Author(s):  
J Satsuma ◽  
K Kajiwara ◽  
B Grammaticos ◽  
J Hietarinta ◽  
A Ramani
Keyword(s):  
Particular Solutions ◽  
Painlevé Ii

scholarly journals The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Pierre Gaillard
Keyword(s):  
Rational Solutions ◽  
Fredholm Determinants ◽  
Infinite Hierarchy

We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x,y) plane in function of time t and parameters a1, a2, b1, and b2.

Sign in / Sign up

Export Citation Format

Share Document