arXiv subjects
Publications and source records attributed to Alexei L. Rebenko.
A new very simple proof of the number of labeled rooted forest-graphs with a given number of vertices is given. As a partial case of this formula we have Cayley's formula.
We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications. Two applications to enumeration of forests with closed-form expressions are given.
In this article, we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive Kirkwood-Salsburg-type equations for the PTCF and write the solutions of these equations as a sum of contributions labelled by certain forests graphs, the connected components of which are tree graphs. We generalize the method introduced by R.A. Minlos and S.K. Poghosyan (1977) in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF which were obtained earlier for lattice spin systems.
We construct the Gibbs state for $ν$-dimensional quantum crystal with site displacements from $\R^d$, $d\geq 1$, and with a one-site \textit{non-polynomial} double-well potential, which has \textit{harmonic} asymptotic growth at infinity. We prove the uniqueness of the corresponding {\it Euclidean Gibbs measure} (EGM) in the \textit{light-mass regime} for the crystal particles. The corresponding state is constructed via a cluster expansion technique for an arbitrary temperature $T\geq 0$. We show that for all $T\geq 0$ the Gibbs state (correlation functions) is analytic with respect to external field conjugated to displacements provided that the mass of particles $m$ is less than a certain value $m_* >0$. The high temperature regime is also discussed.