2014/08/31 by Igor Rumanov
Mathematics · Physics and Astronomy · #Distribution (mathematics) #Generating function #Lax pair #Logarithm #Logarithmic derivative #Nonlinear system #Ode #Ordinary differential equation #Random Matrices and Applications #Series (stratigraphy) #Singularity #Statistical Distribution Estimation and Applications #Statistical Mechanics and Entropy #hep-th #math-ph #math.MP #math.PR #nlin.SI
paper · pdf · doi:10.1007/s00220-015-2487-5
published as Commun. Math. Phys. 342, 843-868 (2016) · presentation improved, references added
openalex publication_date 2015/11/05 · arxiv created 2016/06/08 · arxiv updated 2016/06/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of β. Using this general result, the case β=6 is further considered here. This is the smallest even β, when the corresponding Lax pair and its relation to Painlevé II (PII) have not been known before, unlike cases β=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 β=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 singularity analysis yields series solutions with exponents in the set 4/3, 1/3 and -2/3.