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

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

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

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

The dimension of the SLE curves

Let $\gamma$ be the curve generating a Schramm--Loewner Evolution (SLE) process, with parameter $\kappa\geq0$. We prove that, with probability one, the Hausdorff dimension of $\gamma$ is equal to $\operatorname {Min}(2,1+\kappa/8)$.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

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

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

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 $\mathbb{Z}^2$, 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 $1/2\log(1+\sqrt{2})$. 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