Abstract
A point process is said to be rigid if for any bounded domain in the phase space, the number of particles in the domain is almost surely determined by the restriction of the configuration to the complement of our bounded domain. The main result of this paper is that determinantal point processes with the Airy, the Bessel and the Gamma kernels are rigid. The proof follows the scheme used by Ghosh, Ghosh and Peres: the main step is the construction of a sequence of additive statistics with variance going to zero.
Highlights
1.1 Rigid point processesLet M be a complete separable metric space
Recall that a configuration on M is a purely atomic Radon measure on M; in other words, a collection of particles considered without regard to order and not admitting accumulation points in M
A point process on M is by definition a Borel probability measure on Conf(M)
Summary
Let M be a complete separable metric space. Recall that a configuration on M is a purely atomic Radon measure on M; in other words, a collection of particles considered without regard to order and not admitting accumulation points in M. A point process on M is by definition a Borel probability measure on Conf(M). The following definition of rigidity of a point process is due to Ghosh [6] Definition A point process P on M is called rigid if for any bounded Borel subset. Let μ be a σ -finite Borel probability measure on R, and let (x, y) be the kernel of a locally trace-class operator of orthogonal projection acting in L2(R, μ). Recall that the determinantal point process P is a Borel probability measure on Conf(R) defined by the condition that for any bounded measurable function g, for which g − 1 is supported in a bounded set B, we have. Rigidity of determinantal point processes with the Airy. For the Ginibre ensemble, rigidity has been established by Ghosh and Peres [7]; see Osada and Shirai [16]
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