Provider: Ingenta Connect
Database: Ingenta Connect
Content: application/x-research-info-systems
TY - ABST
AU - Min, Chao
AU - Chen, Yang
TI - Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite n to Double Scaling
JO - Studies in Applied Mathematics
PY - 2018-02-01T00:00:00///
VL - 140
IS - 2
SP - 202
EP - 220
N2 - In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely, the probability that the interval $(-a,a)\phantom{\rule{0.222222em}{0ex}}(0<a<1)$ is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by ${H}_{n}(a)$, ${R}_{n}(a)$, and ${r}_{n}(a)$. We find that each one satisfies a second‐order differential equation. We show that after a double scaling, the large second‐order differential
equation in the variable *a* with *n* as parameter satisfied by ${H}_{n}(a)$ can be reduced to the Jimbo–Miwa–Okamoto σ form of the PainlevĂ© V equation.
UR - https://www.ingentaconnect.com/content/bpl/sapm/2018/00000140/00000002/art00003
M3 - doi:10.1111/sapm.12198
UR - https://doi.org/10.1111/sapm.12198
ER -