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Extreme value distributions of noncolliding diffusion processes

Noncolliding diffusion processes reported in the present paper are $N$-particle systems of diffusion processes in one-dimension, which are conditioned so that all particles start from the origin and never collide with each other in a finite time interval $(0, T)$, $0 < T < \infty$. We consider four temporally inhomogeneous processes with duration $T$, the noncolliding Brownian bridge, the noncolliding Brownian motion, the noncolliding three-dimensional Bessel bridge, and the noncolliding Brownian meander. Their particle distributions at each time $t \in [0, T]$ are related to the eigenvalue distributions of random matrices in Gaussian ensembles and in some two-matrix models. Extreme values of paths in $[0, T]$ are studied for these noncolliding diffusion processes and determinantal and pfaffian representations are given for the distribution functions. The entries of the determinants and pfaffians are expressed using special functions.