1,485 research outputs found
The Max-Flow Min-Cut Theorem for Countable Networks
We prove a strong version of the Max-Flow Min-Cut theorem for countable
networks, namely that in every such network there exist a flow and a cut that
are "orthogonal" to each other, in the sense that the flow saturates the cut
and is zero on the reverse cut. If the network does not contain infinite trails
then this flow can be chosen to be mundane, i.e. to be a sum of flows along
finite paths. We show that in the presence of infinite trails there may be no
orthogonal pair of a cut and a mundane flow. We finally show that for locally
finite networks there is an orthogonal pair of a cut and a flow that satisfies
Kirchhoff's first law also for ends.Comment: 19 pages, to be published in Journal of Combinatorial Theory, Series
A Mechanized Proof of the Max-Flow Min-Cut Theorem for Countable Networks
Aharoni et al. [Ron Aharoni et al., 2010] proved the max-flow min-cut theorem for countable networks, namely that in every countable network with finite edge capacities, there exists a flow and a cut such that the flow saturates all outgoing edges of the cut and is zero on all incoming edges. In this paper, we formalize their proof in Isabelle/HOL and thereby identify and fix several problems with their proof. We also provide a simpler proof for networks where the total outgoing capacity of all vertices other than the source is finite. This proof is based on the max-flow min-cut theorem for finite networks
Random walks in random Dirichlet environment are transient in dimension
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 , RWDE are
parameterized by a -uplet of positive reals. We prove that for all values
of the parameters, RWDE are transient in dimension . 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
.Comment: New version published at PTRF with an analytic proof of lemma
Noise-stability and central limit theorems for effective resistance of random electric networks
We investigate the (generalized) Walsh decomposition of point-to-point
effective resistances on countable random electric networks with i.i.d.
resistances. We show that it is concentrated on low levels, and thus
point-to-point effective resistances are uniformly stable to noise. For graphs
that satisfy some homogeneity property, we show in addition that it is
concentrated on sets of small diameter. As a consequence, we compute the right
order of the variance and prove a central limit theorem for the effective
resistance through the discrete torus of side length in ,
when goes to infinity.Comment: Published at http://dx.doi.org/10.1214/14-AOP996 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Borel circle squaring
We give a completely constructive solution to Tarski's circle squaring
problem. More generally, we prove a Borel version of an equidecomposition
theorem due to Laczkovich. If and are
bounded Borel sets with the same positive Lebesgue measure whose boundaries
have upper Minkowski dimension less than , then and are
equidecomposable by translations using Borel pieces. This answers a question of
Wagon. Our proof uses ideas from the study of flows in graphs, and a recent
result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the
hyperfiniteness of free Borel actions of .Comment: Minor typos correcte
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