302 research outputs found
Hausdorff dimensions for SLE_6
We prove that the Hausdorff dimension of the trace of SLE_6 is almost surely
7/4 and give a more direct derivation of the result (due to
Lawler-Schramm-Werner) that the dimension of its boundary is 4/3. We also prove
that, for all \kappa<8, the SLE_{\kappa} trace has cut-points.Comment: Published at http://dx.doi.org/10.1214/009117904000000072 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Unifying type systems for mobile processes
We present a unifying framework for type systems for process calculi. The
core of the system provides an accurate correspondence between essentially
functional processes and linear logic proofs; fragments of this system
correspond to previously known connections between proofs and processes. We
show how the addition of extra logical axioms can widen the class of typeable
processes in exchange for the loss of some computational properties like
lock-freeness or termination, allowing us to see various well studied systems
(like i/o types, linearity, control) as instances of a general pattern. This
suggests unified methods for extending existing type systems with new features
while staying in a well structured environment and constitutes a step towards
the study of denotational semantics of processes using proof-theoretical
methods
Dessins d'enfants for analysts
We present an algorithmic way of exactly computing Belyi functions for
hypermaps and triangulations in genus 0 or 1, and the associated dessins, based
on a numerical iterative approach initialized from a circle packing combined
with subsequent lattice reduction. The main advantage compared to previous
methods is that it is applicable to much larger graphs; we use very little
algebraic geometry, and aim for this paper to be as self-contained as possible
The dimension of the SLE curves
Let be the curve generating a Schramm--Loewner Evolution (SLE)
process, with parameter . We prove that, with probability one, the
Hausdorff dimension of is equal to .Comment: Published in at http://dx.doi.org/10.1214/07-AOP364 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Quantitative testing semantics for non-interleaving
This paper presents a non-interleaving denotational semantics for the
?-calculus. The basic idea is to define a notion of test where the outcome is
not only whether a given process passes a given test, but also in how many
different ways it can pass it. More abstractly, the set of possible outcomes
for tests forms a semiring, and the set of process interpretations appears as a
module over this semiring, in which basic syntactic constructs are affine
operators. This notion of test leads to a trace semantics in which traces are
partial orders, in the style of Mazurkiewicz traces, extended with readiness
information. Our construction has standard may- and must-testing as special
cases
Percolation without FKG
We prove a Russo-Seymour-Welsh theorem for large and natural perturbative
families of discrete percolation models that do not necessarily satisfy the
Fortuin-Kasteleyn-Ginibre condition of positive association. In particular, we
prove the box-crossing property for the antiferromagnetic Ising model with
small parameter, and for certain discrete Gaussian fields with oscillating
correlation function
On monochromatic arm exponents for 2D critical percolation
We investigate the so-called monochromatic arm exponents for critical
percolation in two dimensions. These exponents, describing the probability of
observing j disjoint macroscopic paths, are shown to exist and to form a
different family from the (now well understood) polychromatic exponents. More
specifically, our main result is that the monochromatic j-arm exponent is
strictly between the polychromatic j-arm and (j+1)-arm exponents.Comment: Published in at http://dx.doi.org/10.1214/10-AOP581 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Smirnov's fermionic observable away from criticality
In a recent and celebrated article, Smirnov [Ann. of Math. (2) 172 (2010)
1435-1467] defines an observable for the self-dual random-cluster model with
cluster weight q = 2 on the square lattice , and uses it to
obtain conformal invariance in the scaling limit. We study this observable away
from the self-dual point. From this, we obtain a new derivation of the fact
that the self-dual and critical points coincide, which implies that the
critical inverse temperature of the Ising model equals .
Moreover, we relate the correlation length of the model to the large deviation
behavior of a certain massive random walk (thus confirming an observation by
Messikh [The surface tension near criticality of the 2d-Ising model (2006)
Preprint]), which allows us to compute it explicitly.Comment: Published in at http://dx.doi.org/10.1214/11-AOP689 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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