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**
Day on Random Spatial Systems
Friday: 8 Februar, 2019, 14:30-18:30 (Universität Augsburg)
**

organized by

Lisa Beck, Dirk Blömker, Nina Gantert (TUM), Sabine Jansen (LMU)

Christoph Hofer-Temmel (NLDA Den Helder and CWI Amsterdam),

Dominic Schuhmacher (Göttingen),

Martin Slowik, (TU Berlin)

(Institute for Mathematics, Room 2004)

14:30 Christoph Hofer-Temmel

15:30 coffee

16:00 Martin Slowik

17:00 Dominic Schuhmacher

**Christoph Hofer-Temmel:
Disagreement percolation for Gibbs point processes**

A Gibbs point process in the low activity (also called high-temperature) regime is expected to behave close to the ideal gas, here the Poisson point process. In particular, one expects just a single Gibbs state, i.e., absence of phase transition, analyticity of the free energy and strong decay of the correlation functions.

Disagreement percolation is a technique to control the differing boundary conditions in a Gibbs specification by a simpler Boolean percolation model. In the low activity regime, the percolation model does not percolate and implies the uniqueness of the Gibbs PP. If the percolation has exponentially decaying connection probabilities, then exponential decay of correlations and analyticity of the free energy holds, too.

This technique has first been extended from the discrete case to bounded range simple Gibbs point processes. An extension to particle processes yields a functional CLT for U-statistics of functionals with finite support. A relaxation of the finite interaction range yields a first proof of uniqueness of the Gibbs state for the continuum random cluster model at low activities. A core building block is a dependent thinning from a Poisson point process to a dominated Gibbs point process within a finite volume, where the thinning probability is related to the derivative of the free energy of the Gibbs point process.

**Dominic Schuhmacher:
Gibbs process approximation in the total variation metric**

Finite Gibbs processes form a flexible class of point processes that have a density with respect to a Poisson process distribution. In this talk we consider Gibbs processes on a compact metric space. We derive an upper bound for the total variation distance between a general Gibbs process distribution and one that satisfies a certain stability condition. The upper bound is expressed in terms of the conditional intensities of the two processes.

At the heart of the proof lies an adaptation of Stein's method to the Gibbs process setting. We re-express the total variation distance in terms of the generator of a spatial birth-death process and reduce the problem to finding an upper bound of the expected coupling time between two spatial birth-death processes with identical transition kernels but started from different configurations.

As applications we consider the total variation distance between two pairwise interaction processes and the Strauss hard core process approximation of an area-interaction process with small interaction parameter.

**Martin Slowik:
Spectral gap estimates and metastable Markov processes**

Metastability is a phenomenon that occurs in the dynamics of a multi-stable non-linear system subject to noise. It is characterized by the existence of multiple, well separated time scales. The talk will be focused on the metastable behavior of reversible Markov chains on countable state spaces. In particular, I will discuss an approach to derive sharp estimates of the constant in the Poincaré inequality. The proof is based on a refined two-scale decomposition. A key ingredient is a discrete analogue of the capacitary inequality of Vladimir Maz'ya leading to estimates that are valid beyond the metastable setting.

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- 23.01.2019