Determinantal Point Processes Associated with Biorthogonal Systems

Abstract

The seven types of noncolliding Brownian bridges constructed so far can be regarded as the time-dependent point processes on the interval $$[0, 2 \pi )$$ or $$[0, \pi ]$$ with time duration [0, T]. Here we define the correlation functions and their generating function called the characteristic function, which specify the point process. In particular, if all correlation functions are expressed by determinants specified by a two-point continuous function, then the point process is said to be determinantal and the two-point function is called the correlation kernel. We prove that our noncolliding Brownian bridges provide seven types of time-dependent determinantal point processes (DPPs) and their correlation kernels are expressed using the biorthogonal $$R_n$$ theta functions. We show that if we take the limit $$t \rightarrow \infty $$ and $$T -t \rightarrow \infty $$ , the seven time-dependent DPPs are reduced to four types of stationary DPPs. Associated with the diffusive scaling consisting of the proper dilatation and time change, we perform the infinite-particle limit $$n \rightarrow \infty $$ . Then we obtain four types of time-dependent DPPs on $$\mathbb {R}$$ or $$\mathbb {R}_+ :=[0, \infty )$$ with an infinite number of particles with time duration [0, T]. Their temporally homogeneous limits are identified with the infinite DPPs well-studied in random matrix theory as the bulk scaling limits of the Gaussian unitary ensemble and its chiral versions. In other words, the present four types of infinite DPPs are elliptic extensions of them.