92,699 research outputs found
Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs
We show how to compute the probabilities of various connection topologies for
uniformly random spanning trees on graphs embedded in surfaces. As an
application, we show how to compute the "intensity" of the loop-erased random
walk in , that is, the probability that the walk from (0,0) to
infinity passes through a given vertex or edge. For example, the probability
that it passes through (1,0) is 5/16; this confirms a conjecture from 1994
about the stationary sandpile density on . We do the analogous
computation for the triangular lattice, honeycomb lattice and , for which the probabilities are 5/18, 13/36, and
respectively.Comment: 45 pages, many figures. v2 has an expanded introduction, a revised
section on the LERW intensity, and an expanded appendix on the annular matri
Management of Technology Focused on the Water Analysis Results in Artesians Wells
The water is an universal soluble, fundamental to every living being. The main component to the human body and indispensable for any form of life, however, there is an increasing preocupation within the quality of their providers, which are the rivers, strands and springs, but as the time passes by are threatened by anthropogenic activities, main causer of the contamination and destruction of the local fauna and habitat. The principal objetictive of this research was to generate necessary information to further make use of this water, specifically located in Porto Velho, Rondônia, in Jamari\u27s River\u27s hidrographic basin sided with Green River\u27s, one affluent and one subfluent of Madeira\u27s River, which is one of the most important hidrographic basins of Amazon\u27s River, yet, lots of physical, chemicals and microbiological parameters were effected, like pH, turbidity, overall alkalinity, overall toughness, iron, chloride, color, fecal coliforms, that are capable of identify the contamination by anthropic action, obtaining the characteristics of the containing waters in the studied area, which in turn had favorable results, following the collected results we could confirm that all them were under acceptable and expectable parameters required by legislation in force of bathing, aquatic community preservation and human consumption, being the last one dependable of a simple threatment (chlorination)
Products of Independent Non-Hermitian Random Matrices
For fixed , we consider independent non-Hermitian
random matrices with i.i.d. centered entries with a finite
-th moment, As tends to infinity, we show that the
empirical spectral distribution of n^{-m/2} \*X_1 X_2 ... X_m converges, with
probability 1, to a non-random, rotationally invariant distribution with
compact support in the complex plane. The limiting distribution is the -th
power of the circular law.Comment: The paper is accepted for publication in Electronic Journal of
Probability. In the final version we fixed a small technical gap in the
proofs of Theorem 15 and Lemma 19 and added the reference to the 2009 paper
by Girko and Vladimirov
On the Complexity of Digraph Colourings and Vertex Arboricity
It has been shown by Bokal et al. that deciding 2-colourability of digraphs
is an NP-complete problem. This result was later on extended by Feder et al. to
prove that deciding whether a digraph has a circular -colouring is
NP-complete for all rational . In this paper, we consider the complexity
of corresponding decision problems for related notions of fractional colourings
for digraphs and graphs, including the star dichromatic number, the fractional
dichromatic number and the circular vertex arboricity. We prove the following
results:
Deciding if the star dichromatic number of a digraph is at most is
NP-complete for every rational .
Deciding if the fractional dichromatic number of a digraph is at most is
NP-complete for every .
Deciding if the circular vertex arboricity of a graph is at most is
NP-complete for every rational .
To show these results, different techniques are required in each case. In
order to prove the first result, we relate the star dichromatic number to a new
notion of homomorphisms between digraphs, called circular homomorphisms, which
might be of independent interest. We provide a classification of the
computational complexities of the corresponding homomorphism colouring problems
similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur
Spectrum of Markov generators on sparse random graphs
Correction in Proposition 4.3. Final version.International audienceWe investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if kj and L(j,j)=-sum(L(j,k),kj), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős-Rényi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L
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