On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation. (English) Zbl 1450.34066
In this paper, the authors study a one parameter family of tronquée solutions of the second Painlevé equation \(q^{\prime \prime}=2q^3+sq-\nu\). By the Riemann-Hilbert correspondence, the solution space of the second Painlevé equation is represented by a cubic
\[s_1 -s_2 +s_3 +s_1s_2s_3 = -2 \sin ( \nu \pi).\]
The tronquée solution corresponds to the case \(s_1 = 1/s_3=- e^{-2 \alpha \pi i},\) where \( \nu =2 \alpha +1/2\).
Such tronquée solutions contain the generalized Hastings-McLeod solution and the Airy-type solutions. The Tracy-Widom distribution is defined by an integral of the Hastings-McLeod solution (\(\nu =0\)). In this paper, asymptotics of an integral of the tronquée solutions are described when \( \alpha > -1/2\) and \(s_2 >0.\) Since the Hamiltonian of the tronquée solutions is pole-free on the real line, a regularized integral of the Hamiltonian admits an asymptotic expansions represented by the Riemann zeta-function and Barnes’ \(G\)-function. Such results have applications to a perturbed Gaussian unitary ensemble and a concrete unitary random matrix ensemble in random matrix theory.
Such tronquée solutions contain the generalized Hastings-McLeod solution and the Airy-type solutions. The Tracy-Widom distribution is defined by an integral of the Hastings-McLeod solution (\(\nu =0\)). In this paper, asymptotics of an integral of the tronquée solutions are described when \( \alpha > -1/2\) and \(s_2 >0.\) Since the Hamiltonian of the tronquée solutions is pole-free on the real line, a regularized integral of the Hamiltonian admits an asymptotic expansions represented by the Riemann zeta-function and Barnes’ \(G\)-function. Such results have applications to a perturbed Gaussian unitary ensemble and a concrete unitary random matrix ensemble in random matrix theory.
Reviewer: Yousuke Ohyama (Tokushima)
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 33E17 | Painlevé-type functions |
| 60B20 | Random matrices (probabilistic aspects) |
Software:
DLMFReferences:
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