5,778 research outputs found
Painting a graph with competing random walks
Let be independent random walks on , , each
starting from the uniform distribution. Initially, each site of
is unmarked, and, whenever visits such a site, it is set
irreversibly to . The mean of , the cardinality of the set
of sites painted by , once all of has been
visited, is by symmetry. We prove the following conjecture due
to Pemantle and Peres: for each there exists a constant
such that where ,
and for . We will also identify explicitly and
show that as . This is a special case of a more
general theorem which gives the asymptotics of
for a large class of transient, vertex
transitive graphs; other examples include the hypercube and the Caley graph of
the symmetric group generated by transpositions.Comment: Published in at http://dx.doi.org/10.1214/11-AOP713 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Quantum Loewner Evolution
What is the scaling limit of diffusion limited aggregation (DLA) in the
plane? This is an old and famously difficult question. One can generalize the
question in two ways: first, one may consider the {\em dielectric breakdown
model} -DBM, a generalization of DLA in which particle locations are
sampled from the -th power of harmonic measure, instead of harmonic
measure itself. Second, instead of restricting attention to deterministic
lattices, one may consider -DBM on random graphs known or believed to
converge in law to a Liouville quantum gravity (LQG) surface with parameter
.
In this generality, we propose a scaling limit candidate called quantum
Loewner evolution, QLE. QLE is defined in terms of the radial
Loewner equation like radial SLE, except that it is driven by a measure valued
diffusion derived from LQG rather than a multiple of a standard
Brownian motion. We formalize the dynamics of using an SPDE. For each
, there are two or three special values of for which
we establish the existence of a solution to these dynamics and explicitly
describe the stationary law of .
We also explain discrete versions of our construction that relate DLA to
loop-erased random walk and the Eden model to percolation. A certain
"reshuffling" trick (in which concentric annular regions are rotated randomly,
like slot machine reels) facilitates explicit calculation.
We propose QLE as a scaling limit for DLA on a random
spanning-tree-decorated planar map, and QLE as a scaling limit for the
Eden model on a random triangulation. We propose using QLE to endow
pure LQG with a distance function, by interpreting the region explored by a
branching variant of QLE, up to a fixed time, as a metric ball in a
random metric space.Comment: 132 pages, approximately 100 figures and computer simulation
Intersections of SLE Paths: the double and cut point dimension of SLE
We compute the almost-sure Hausdorff dimension of the double points of
chordal SLE_kappa for kappa > 4, confirming a prediction of Duplantier-Saleur
(1989) for the contours of the FK model. We also compute the dimension of the
cut points of chordal SLE_kappa for kappa > 4 as well as analogous dimensions
for the radial and whole-plane SLE_kappa(rho) processes for kappa > 0. We
derive these facts as consequences of a more general result in which we compute
the dimension of the intersection of two flow lines of the formal vector field
e^{ih/chi}, where h is a Gaussian free field and chi > 0, of different angles
with each other and with the domain boundary.Comment: 70 page, 26 figure
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