688 research outputs found
Phase Transitions for Random Walk Asymptotics on Free Products of Groups
Suppose we are given finitely generated groups
equipped with irreducible random walks. Thereby we assume that the expansions
of the corresponding Green functions at their radii of convergence contain only
logarithmic or algebraic terms as singular terms up to sufficiently large order
(except for some degenerate cases). We consider transient random walks on the
free product {} and give a complete
classification of the possible asymptotic behaviour of the corresponding
-step return probabilities. They either inherit a law of the form
from one of the free
factors or obey a -law, where
is the corresponding spectral radius and is the period of
the random walk. In addition, we determine the full range of the asymptotic
behaviour in the case of nearest neighbour random walks on free products of the
form . Moreover, we characterize the possible
phase transitions of the non-exponential types
in the case .Comment: 32 page
Exotic local limit theorems at the phase transition in free products
We construct random walks on free products of the form Z 3 * Z d , with d = 5
or 6 which are divergent and not spectrally positive recurrent. We then derive
a local limit theorem for these random walks, proving that * n (e)
CR --n n --5/3 if d = 5 and * n (e) CR --n n --3/2 log(n) --1/2 if
d = 6, where * n is the nth convolution power of and R is the
inverse of the spectral radius of . This disproves a result of Candellero
and Gilch [7] and a result of the authors of this paper that was stated in a
rst version of [11]. This also shows that the classication of local limit
theorems on free products of the form Z d 1 * Z d 2 or more generally on
relatively hyperbolic groups with respect to virtually abelian subgroups is
incomplete
One-dimensional quantum walks with one defect
The CGMV method allows for the general discussion of localization properties
for the states of a one-dimensional quantum walk, both in the case of the
integers and in the case of the non negative integers. Using this method we
classify, according to such localization properties, all the quantum walks with
one defect at the origin, providing explicit expressions for the asymptotic
return probabilities at the origin
Branching Random Walks on Free Products of Groups
We study certain phase transitions of branching random walks (BRW) on Cayley
graphs of free products. The aim of this paper is to compare the size and
structural properties of the trace, i.e., the subgraph that consists of all
edges and vertices that were visited by some particle, with those of the
original Cayley graph. We investigate the phase when the growth parameter
is small enough such that the process survives but the trace is not
the original graph. A first result is that the box-counting dimension of the
boundary of the trace exists, is almost surely constant and equals the
Hausdorff dimension which we denote by . The main result states
that the function has only one point of discontinuity which is
at where is the radius of convergence of the Green function
of the underlying random walk. Furthermore, is bounded by one half
the Hausdorff dimension of the boundary of the original Cayley graph and the
behaviour of as is classified.
In the case of free products of infinite groups the end-boundary can be
decomposed into words of finite and words of infinite length. We prove the
existence of a phase transition such that if
the end boundary of the trace consists only of infinite words and if
it also contains finite words. In the last case,
the Hausdorff dimension of the set of ends (of the trace and the original
graph) induced by finite words is strictly smaller than the one of the ends
induced by infinite words.Comment: 39 pages, 4 figures; final version, accepted for publication in the
Proceedings of LM
Convex hulls of random walks, hyperplane arrangements, and Weyl chambers
We give an explicit formula for the probability that the convex hull of an n-step random walk in Rd does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (Skand Aktuarietidskr 32:27–36, 1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments.
This probabilistic problem is shown to be equivalent to either of the two geometric ones: (1) Find the number of Weyl chambers of type Bn intersected by a generic linear subspace of Rn of codimension d; (2) Find the conic intrinsic volumes of a Weyl chamber of type Bn. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (Discrete Comput Geom 46(3):417–426, 2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement.
We obtain analogous distribution-free results for Weyl chambers of type An−1 (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type Dn, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form B1 ×···× B1 recovers the Wendel formula (Math Scand 11:109–111, 1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin.
We also give an asymptotic analysis of the obtained absorption probabilities as n → ∞, in both cases of fixed and increasing dimension d
Localization in disordered superconducting wires with broken spin-rotation symmetry
Localization and delocalization of non-interacting quasiparticle states in a
superconducting wire are reconsidered, for the cases in which spin-rotation
symmetry is absent, and time-reversal symmetry is either broken or unbroken;
these are referred to as symmetry classes BD and DIII, respectively. We show
that, if a continuum limit is taken to obtain a Fokker-Planck (FP) equation for
the transfer matrix, as in some previous work, then when there are more than
two scattering channels, all terms that break a certain symmetry are lost. It
was already known that the resulting FP equation exhibits critical behavior.
The additional symmetry is not required by the definition of the symmetry
classes; terms that break it arise from non-Gaussian probability distributions,
and may be kept in a generalized FP equation. We show that they lead to
localization in a long wire. When the wire has more than two scattering
channels, these terms are irrelevant at the short distance (diffusive or
ballistic) fixed point, but as they are relevant at the long-distance critical
fixed point, they are termed dangerously irrelevant. We confirm the results in
a supersymmetry approach for class BD, where the additional terms correspond to
jumps between the two components of the sigma model target space. We consider
the effect of random fluxes, which prevent the system localizing. We show
that in one dimension the transitions in these two symmetry classes, and also
those in the three chiral symmetry classes, all lie in the same universality
class
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