We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self-dual pointThis gives a proof that the critical temperature of the q-state Potts model is equal to log(1 + √ q) for all q 2. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all q 1, in contrast to earlier methods valid only for certain given q. The proof extends to the triangular and the hexagonal lattices as well. Mathematics Subject Classification (2000)60K35 · 82B20 (primary); 82B26 · 82B43

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

International audienceWe prove a Russo-Seymour-Welsch percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let $U$ be a smooth connected bounded open set in $\mathbb R^2$ and $\gamma, \gamma'$ two disjoint arcs of positive length in the boundary of $U$. We prove that there exists a positive constant $c$, such that for any positive scale $s$, with probability at least $c$ there exists a connected component of $\{x\in \bar U, \, f(sx) \textgreater{} 0\} $ intersecting both $\gamma$ and $\gamma'$, where $f$ is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For $s$ large enough, the same conclusion holds for the zero set $\{x\in \bar U, \, f(sx) = 0\} $. As an important intermediate result, we prove that sign percolation for a general stationary Gaussian field can be made equivalent to a correlated percolation model on a lattice

We prove that the Hausdorff dimension of the trace of SLE6 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 κ < 8, the SLEκ trace has cut-points.