7,758 research outputs found
The hyperbolic geometry of random transpositions
Turn the set of permutations of objects into a graph by connecting
two permutations that differ by one transposition, and let be the
simple random walk on this graph. In a previous paper, Berestycki and Durrett
[In Discrete Random Walks (2005) 17--26] showed that the limiting behavior of
the distance from the identity at time has a phase transition at .
Here we investigate some consequences of this result for the geometry of .
Our first result can be interpreted as a breakdown for the Gromov hyperbolicity
of the graph as seen by the random walk, which occurs at a critical radius
equal to . Let be a triangle formed by the origin and two points
sampled independently from the hitting distribution on the sphere of radius
for a constant . Then when , if the geodesics are suitably
chosen, with high probability is -thin for some , whereas
it is always O(n)-thick when . We also show that the hitting
distribution of the sphere of radius is asymptotically singular with
respect to the uniform distribution. Finally, we prove that the critical
behavior of this Gromov-like hyperbolicity constant persists if the two
endpoints are sampled from the uniform measure on the sphere of radius .
However, in this case, the critical radius is .Comment: Published at http://dx.doi.org/10.1214/009117906000000043 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Two-dimensional gauge theories of the symmetric group S(n) and branched n-coverings of Riemann surfaces in the large-n limit
Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge
theory of the symmetric group S(n) defined on a cell discretization of the
surface. We study the theory in the large-n limit, and we find a rich phase
diagram with first and second order transition lines. The various phases are
characterized by different connectivity properties of the covering surface. We
point out some interesting connections with the theory of random walks on group
manifolds and with random graph theory.Comment: Talk presented at the "Light-cone physics: particles and strings",
Trento, Italy, September 200
Effect of scale on long-range random graphs and chromosomal inversions
We consider bond percolation on vertices on a circle where edges are
permitted between vertices whose spacing is at most some number L=L(n). We show
that the resulting random graph gets a giant component when
(when the mean degree exceeds 1) but not when . The proof uses
comparisons to branching random walks. We also consider a related process of
random transpositions of particles on a circle, where transpositions only
occur again if the spacing is at most . Then the process exhibits the
mean-field behavior described by Berestycki and Durrett if and only if L(n)
tends to infinity, no matter how slowly. Thus there are regimes where the
random graph has no giant component but the random walk nevertheless has a
phase transition. We discuss possible relevance of these results for a dataset
coming from D. repleta and D. melanogaster and for the typical length of
chromosomal inversions.Comment: Published in at http://dx.doi.org/10.1214/11-AAP793 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Mixing time of Metropolis chain based on random transposition walk converging to multivariate Ewens distribution
We prove sharp rates of convergence to the Ewens equilibrium distribution for
a family of Metropolis algorithms based on the random transposition shuffle on
the symmetric group, with starting point at the identity. The proofs rely
heavily on the theory of symmetric Jack polynomials, developed initially by
Jack [Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971) 1-18], Macdonald
[Symmetric Functions and Hall Polynomials (1995) New York] and Stanley [Adv.
Math. 77 (1989) 76-115]. This completes the analysis started by Diaconis and
Hanlon in [Contemp. Math. 138 (1992) 99-117]. In the end we also explore other
integrable Markov chains that can be obtained from symmetric function theory.Comment: Published at http://dx.doi.org/10.1214/14-AAP1031 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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