On graph equivalences preserved under extensions

Let R be an equivalence relation on graphs. By the strengthening of R we mean the relation R' such that graphs G and H are in the relation R' if for every graph F, the union of the graphs G and F is in the relation R with the union of the graphs H and F. We study strengthenings of equivalence relations on graphs. The most important case that we consider concerns equivalence relations defined by graph properties. We obtain results on the strengthening of equivalence relations determined by the properties such as being a k-connected graph, k-colorable, hamiltonian and planar

Maximal stream and minimal cutset for first passage percolation through a domain of Rd\mathbb{R}^d

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq2$ and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb{R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint open subsets of $\Gamma$, representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. A law of large numbers for the maximal flow from $\Gamma ^1$ to $\Gamma ^2$ in $\Omega$ is already known. In this paper we investigate the asymptotic behavior of a maximal stream and a minimal cutset. A maximal stream is a vector measure $\vec{\mu}_n^{\max}$ that describes how the maximal amount of fluid can cross $\Omega$. Under conditions on the regularity of the domain and on the law of the capacities of the edges, we prove that the sequence $(\vec{\mu}_n^{\max})_{n\geq1}$ converges a.s. to the set of the solutions of a continuous deterministic problem of maximal stream in an anisotropic network. A minimal cutset can been seen as the boundary of a set $E_n^{\min}$ that separates $\Gamma ^1$ from $\Gamma ^2$ in $\Omega$ and whose random capacity is minimal. Under the same conditions, we prove that the sequence $(E_n^{\min})_{n\geq1}$ converges toward the set of the solutions of a continuous deterministic problem of minimal cutset. We deduce from this a continuous deterministic max-flow min-cut theorem and a new proof of the law of large numbers for the maximal flow. This proof is more natural than the existing one, since it relies on the study of maximal streams and minimal cutsets, which are the pertinent objects to look at.Comment: Published in at http://dx.doi.org/10.1214/13-AOP851 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Antichain cutsets of strongly connected posets

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we get such a characterization for semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We also consider a generalization to strongly connected hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio

Random walks in random Dirichlet environment are transient in dimension d≥3d\ge 3

We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $\Z^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $d\ge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $d\ge 3$.Comment: New version published at PTRF with an analytic proof of lemma