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Multi angle  Anchored expansion in the hyperbolic Poisson Voronoi tessellation
Paquette, Elliot (Auteur de la Conférence) | CIRM (Editeur )

We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tessellation. We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive ...

05C80 ; 60D05 ; 60G55

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Post-edited  Freezing and decorated Poisson point processes
Zeitouni, Ofer (Auteur de la Conférence) | CIRM (Editeur )

The freezing in the title refers to a property of point processes: let $\left ( X_i \right )_{i\geq 1}$ denote a point process which is locally finite and has finite maximum. For a function f continuous of compact support, define $Z_f=f\left ( X_1 \right )+f\left ( X_2 \right )+....$ We say that freezing occurs if the Laplace transform of $Z_f$ depends on f only through a shift. I will discuss this notion and its equivalence with other properties of the point process. In particular, such freezing occurs for the extremal process in branching random walks and in certain versions of the (discrete) two dimensional GFF.
Joint work with Eliran Subag
The freezing in the title refers to a property of point processes: let $\left ( X_i \right )_{i\geq 1}$ denote a point process which is locally finite and has finite maximum. For a function f continuous of compact support, define $Z_f=f\left ( X_1 \right )+f\left ( X_2 \right )+....$ We say that freezing occurs if the Laplace transform of $Z_f$ depends on f only through a shift. I will discuss this notion and its equivalence with other ...

60G55 ; 60J65 ; 60J80

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Multi angle  The Poisson-saddlepoint approximation
Baddeley, Adrian (Auteur de la Conférence) | CIRM (Editeur )

Gibbs spatial point processes are important models in theoretical physics and in spatial statistics. After a brief survey of Gibbs point processes, we will present a method for approximating their most important characteristic, the intensity of the process. The method has some affinity with the classical saddlepoint approximations of probability densities. For pairwise-interaction processes the approximation can be computed directly : it performs very well in many cases, but not in all cases. For higher-order interactions, we invoke limit results from stochastic geometry due to Roger Miles and the late Peter Hall, in order to compute the approximation.

Joint work with Gopalan Nair.
Gibbs spatial point processes are important models in theoretical physics and in spatial statistics. After a brief survey of Gibbs point processes, we will present a method for approximating their most important characteristic, the intensity of the process. The method has some affinity with the classical saddlepoint approximations of probability densities. For pairwise-interaction processes the approximation can be computed directly : it ...

60G55 ; 82B21 ; 62E17

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Multi angle  A functional limit theorem for the sine-process
Dymov, Andrey (Auteur de la Conférence) | CIRM (Editeur )

It is well-known that a large class of determinantal processes including the sine-process satisfies the Central Limit Theorem. For many dynamical systems satisfying the CLT the Donsker Invariance Principle also takes place. The latter states that, in some appropriate sense, trajectories of the system can be approximated by trajectories of the Brownian motion. I will present results of my joint work with A. Bufetov, where we prove a functional limit theorem for the sine-process, which turns out to be very different from the Donsker Invariance Principle. We show that the anti-derivative of our process can be approximated by the sum of a linear Gaussian process and small independent Gaussian fluctuations whose covariance matrix we compute explicitly. It is well-known that a large class of determinantal processes including the sine-process satisfies the Central Limit Theorem. For many dynamical systems satisfying the CLT the Donsker Invariance Principle also takes place. The latter states that, in some appropriate sense, trajectories of the system can be approximated by trajectories of the Brownian motion. I will present results of my joint work with A. Bufetov, where we prove a functional ...

60G55 ; 60F05 ; 60G60

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Multi angle  Conditioned determinantal processes are determinantal
Shamov, Alexander (Auteur de la Conférence) | CIRM (Editeur )

A determinantal point process governed by a Hermitian contraction kernel $K$ on a measure space $E$ remains determinantal when conditioned on its configuration on a subset $B \subset E$. Moreover, the conditional kernel can be chosen canonically in a way that is "local" in a non-commutative sense, i.e. invariant under "restriction" to closed subspaces $L^2(B) \subset P \subset L^2(E)$. Using the properties of the canonical conditional kernel we establish a conjecture of Lyons and Peres: if $K$ is a projection then almost surely all functions in its image can be recovered by sampling at the points of the process.
Joint work with Alexander Bufetov and Yanqi Qiu.
A determinantal point process governed by a Hermitian contraction kernel $K$ on a measure space $E$ remains determinantal when conditioned on its configuration on a subset $B \subset E$. Moreover, the conditional kernel can be chosen canonically in a way that is "local" in a non-commutative sense, i.e. invariant under "restriction" to closed subspaces $L^2(B) \subset P \subset L^2(E)$. Using the properties of the canonical conditional kernel ...

60G55 ; 60C05

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Multi angle  The Airy point process in the two-periodic Aztec diamond
Johansson, Kurt (Auteur de la Conférence) | CIRM (Editeur )

The two-periodic Aztec diamond is a dimer or random tiling model with three phases, solid, liquid and gas. The dimers form a determinantal point process with a somewhat complicated but explicit correlation kernel. I will discuss in some detail how the Airy point process can be found at the liquid-gas boundary by looking at suitable averages of height function differences. The argument is a rather complicated analysis using the cumulant approach and subtle cancellations. Joint work with Vincent Beffara and Sunil Chhita. The two-periodic Aztec diamond is a dimer or random tiling model with three phases, solid, liquid and gas. The dimers form a determinantal point process with a somewhat complicated but explicit correlation kernel. I will discuss in some detail how the Airy point process can be found at the liquid-gas boundary by looking at suitable averages of height function differences. The argument is a rather complicated analysis using the cumulant approach ...

60K35 ; 60G55 ; 60C05 ; 82B20 ; 05B45

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Multi angle  Fractional Poisson process: long-range dependence and applications in ruin theory
Biard, Romain (Auteur de la Conférence) | CIRM (Editeur )

We study a renewal risk model in which the surplus process of the insurance company is modeled by a compound fractional Poisson process. We establish the long-range dependence property of this non-stationary process. Some results for the ruin probabilities are presented in various assumptions on the distribution of the claim sizes.

60G22 ; 60G55 ; 91B30 ; 60K05 ; 33E12