385,112 research outputs found
Percolation in random environment
We consider bond percolation on the square lattice with perfectly correlated
random probabilities. According to scaling considerations, mapping to a random
walk problem and the results of Monte Carlo simulations the critical behavior
of the system with varying degree of disorder is governed by new, random fixed
points with anisotropic scaling properties. For weaker disorder both the
magnetization and the anisotropy exponents are non-universal, whereas for
strong enough disorder the system scales into an {\it infinite randomness fixed
point} in which the critical exponents are exactly known.Comment: 8 pages, 7 figure
Statistical Models on Spherical Geometries
We use a one-dimensional random walk on -dimensional hyper-spheres to
determine the critical behavior of statistical systems in hyper-spherical
geometries. First, we demonstrate the properties of such a walk by studying the
phase diagram of a percolation problem. We find a line of second and first
order phase transitions separated by a tricritical point. Then, we analyze the
adsorption-desorption transition for a polymer growing near the attractive
boundary of a cylindrical cell membrane. We find that the fraction of adsorbed
monomers on the boundary vanishes exponentially when the adsorption energy
decreases towards its critical value.Comment: 8 pages, latex, 2 figures in p
Sliding blocks with random friction and absorbing random walks
With the purpose of explaining recent experimental findings, we study the
distribution of distances traversed by a block that
slides on an inclined plane and stops due to friction. A simple model in which
the friction coefficient is a random function of position is considered.
The problem of finding is equivalent to a First-Passage-Time
problem for a one-dimensional random walk with nonzero drift, whose exact
solution is well-known. From the exact solution of this problem we conclude
that: a) for inclination angles less than \theta_c=\tan(\av{\mu})
the average traversed distance \av{\lambda} is finite, and diverges when
as \av{\lambda} \sim (\theta_c-\theta)^{-1}; b) at
the critical angle a power-law distribution of slidings is obtained:
. Our analytical results are confirmed by
numerical simulation, and are in partial agreement with the reported
experimental results. We discuss the possible reasons for the remaining
discrepancies.Comment: 8 pages, 8 figures, submitted to Phys. Rev.
A.S. convergence for infinite colour P\'olya urns associated with stable random walks
We answer Problem 11.1 of Janson arXiv:1803.04207 on P\'olya urns associated
with stable random walk. Our proof use neither martingales nor trees, but an
approximation with a differential equation.Comment: 8 page
Irreducible compositions and the first return to the origin of a random walk
Let be a pair of compositions of
into positive parts. We say this pair is {\em irreducible} if there is
no positive for which . The
probability that a random pair of compositions of is irreducible is shown
to be asymptotic to . This problem leads to a problem in probability
theory. Two players move along a game board by rolling a die, and we ask when
the two players will first coincide. A natural extension is to show that the
probability of a first return to the origin at time for any mean-zero
variance random walk is asymptotic to . We prove
this via two methods, one analytic and one probabilistic
Critical behavior of the contact process in annealed scale-free networks
Critical behavior of the contact process is studied in annealed scale-free
networks by mapping it on the random walk problem. We obtain the analytic
results for the critical scaling, using the event-driven dynamics approach.
These results are confirmed by numerical simulations. The disorder fluctuation
induced by the sampling disorder in annealed networks is also explored.
Finally, we discuss over the discrepancy of the finite-size-scaling theory in
annealed and quenched networks in spirit of the droplet size scale and the
linking disorder fluctuation.Comment: 8 pages, 5 figure
Random Walks on a Fluctuating Lattice: A Renormalization Group Approach Applied in One Dimension
We study the problem of a random walk on a lattice in which bonds connecting
nearest neighbor sites open and close randomly in time, a situation often
encountered in fluctuating media. We present a simple renormalization group
technique to solve for the effective diffusive behavior at long times. For
one-dimensional lattices we obtain better quantitative agreement with
simulation data than earlier effective medium results. Our technique works in
principle in any dimension, although the amount of computation required rises
with dimensionality of the lattice.Comment: PostScript file including 2 figures, total 15 pages, 8 other figures
obtainable by mail from D.L. Stei
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