Gibbs cluster measures on configuration spaces

The probability distribution g_cl of a Gibbs cluster point process in X = R^d (with i.i.d. random clusters attached to points of a Gibbs configuration with distribution g) is studied via the projection of an auxiliary Gibbs measure ĝ in the space of configurations ^γ={(x,\bar{y})}, where x∈X indicates a cluster "center" and y∈\mathfrak{X}=\sqcup_{n} X^n represents a corresponding cluster relative to x. We show that the measure g_cl is quasi-invariant with respect to the group Diff_0(X) of compactly supported diffeomorphisms of X, and prove an integration-by-parts formula for g_cl. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms

Poisson cluster measures : Quasi-invariance, integration by parts and equilibrium stochastic dynamics

The distribution µcl of a Poisson cluster process in X = Rd (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X = FnXn, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure µcl is quasiinvariant with respect to the group of compactly supported diffeomorphisms ofX and prove an integration-by-parts formula for µcl. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet form

Universality of the limit shape of convex lattice polygonal lines

Let ${\varPi}_n$ be the set of convex polygonal lines $\varGamma$ with vertices on $\mathbb {Z}_+^2$ and fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit shape, as $n\to\infty$, of "typical" $\varGamma\in {\varPi}_n$ with respect to a parametric family of probability measures $\{P_n^r,0<r<\infty\}$ on ${\varPi}_n$, including the uniform distribution ($r=1$) for which the limit shape was found in the early 1990s independently by A. M. Vershik, I. B\'ar\'any and Ya. G. Sinai. We show that, in fact, the limit shape is universal in the class $\{P^r_n\}$, even though $P^r_n$ ($r\ne1$) and $P^1_n$ are asymptotically singular. Measures $P^r_n$ are constructed, following Sinai's approach, as conditional distributions $Q_z^r(\cdot |{\varPi}_n)$, where $Q_z^r$ are suitable product measures on the space ${\varPi}=\bigcup_n{\varPi}_n$, depending on an auxiliary "free" parameter $z=(z_1,z_2)$. The transition from $({\varPi},Q_z^r)$ to $({\varPi}_n,P_n^r)$ is based on the asymptotics of the probability $Q_z^r({\varPi}_n)$, furnished by a certain two-dimensional local limit theorem. The proofs involve subtle analytical tools including the M\"obius inversion formula and properties of zeroes of the Riemann zeta function.Comment: Published in at http://dx.doi.org/10.1214/10-AOP607 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org