arXiv subjects

Eugene A. Pechersky

Publications and source records attributed to Eugene A. Pechersky.

2 recordsLinked to original sources

arXiv2010

Gibbs Random Graphs

Consider a discrete locally finite subset $Γ$ of $R^d$ and the complete graph $(Γ,E)$, with vertices $Γ$ and edges $E$. We consider Gibbs measures on the set of sub-graphs with vertices $Γ$ and edges $E'\subset E$. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when the $Γ$ is a sample from homogeneous Poisson process and (b) for a fixed $Γ$ with exponential decay of connectivity.

math.PR↗

arXiv2007

Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on $\mathbb{R}^{\mathbb{Z}^{d}}$ with Hamiltonian $H(\x)=\sum_{i,j}J_{i,j}(\x(i)-\x(j))^{2}+ h\sum_{i}(\x(i)-\dd(i))^{2}$, where $\x, \dd$ are configurations, $\x(i),\dd(i)\in\mathbb{R}$, $i,j\in{\mathbb{Z}^{d}}$. The configuration $\dd$ is given and considered as observations. The `couplings' $J_{i,j}$ are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration $\m$ corresponding to the Hamiltonian.

math.PR↗