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Register a new userConditional measures for Pfaffian point processes: conditioning on a bounded domain
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For a Pfaffian point process we show that its Palm measures, its normalised compositions with multiplicative functionals, and its conditional measures with respect to fixing the configuration in a bounded subset are Pfaffian point processes whose kernels we find explicitly.
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Our first result states that the orthogonal and symplectic Bessel processes are rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceeds by an estimate of the variance of additive statistics in the spirit of Ghosh and Peres. Second, a sufficient condition for number rigidity of stationary Pfaffian processes, relying on the Kolmogorov criterion for interpolation of stationary processes and applicable, in particular, to pfaffian sine-processes, is given in terms of the asymptotics of the spectral measure for additive statistics.
The gamma kernels are a family of projection kernels $K^{(z,z)}=K^{(z,z)}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Eulers gamma function and depend on two continuous parameters $z,z$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z)}$ serves as a correlation kernel for a determinantal measure $M^{(z,z)}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$ ldots, K^{(z-1,z-1)}, ; K^{(z,z)},; K^{(z+1,z+1)}, ldots, $$ and establish the following hierarchical relations inside any such chain: Given $(z,z)$, the kernel $K^{(z,z)}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z+1)}$, and the one-point Palm distributions for the measure $M^{(z,z)}$ are absolutely continuous with respect to $M^{(z+1,z+1)}$. We also explicitly compute the corresponding Radon-Nikodym derivatives and show that they are given by certain normalized multiplicative functionals.
For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, omega)$ over a domain $U subset mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^infty(U)$-module structure of $A^2(U, omega)$. A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of $U$.
We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the A-power of the jump length and depend on the energy marks via a Boltzmann--like factor. The case A=1 corresponds to the phonon-induced Mott variable range hopping in disordered solids in the regime of strong Anderson localization. We prove that for almost every realization of the marked process, the diffusively rescaled random walk, with arbitrary start point, converges to a Brownian motion whose diffusion matrix is positive definite, and independent of the environment. Finally, we extend the above result to other point processes including diluted lattices.
We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $l$ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $l$ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $l$ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among other things that a phase transition occurs at c = 14/15 for the connectedness of the loops that touch $l$. Our results can be viewed as an extension of some of the results in our earlier paper in the following two directions: There, a loop-soup cluster was conditioned on its entire outer boundary while we discover here only part of this boundary. And, while it was explained there that the strong decomposition using a Poisson point process of excursions that we derived there should be specific to the case of the critical loop-soup, we show here that in the subcritical cases, a weaker property involving the conformal restriction property nevertheless holds.
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