TY - JOUR
T1 - Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System
AU - Xu, Shuai-Xia
AU - Dai, Dan
PY - 2019/1
Y1 - 2019/1
N2 - We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy–Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.
AB - We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy–Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.
UR - http://www.scopus.com/inward/record.url?scp=85053451192&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85053451192&origin=recordpage
U2 - 10.1007/s00220-018-3257-y
DO - 10.1007/s00220-018-3257-y
M3 - 21_Publication in refereed journal
VL - 365
SP - 515
EP - 567
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 2
ER -