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Register a new userPlanar random-cluster model: fractal properties of the critical phase
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This paper is studying the critical regime of the planar random-cluster model on $mathbb Z^2$ with cluster-weight $qin[1,4)$. More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on said boundary. Additionally, they imply that any sub-sequential scaling limit of the collection of interfaces between primal and dual clusters is made of loops that are non-simple. We also obtain a number of properties of so-called arm-events: three universal critical exponents (two arms in the half-plane, three arms in the half-plane and five arms in the bulk), quasi-multiplicativity and well-separation properties (even when arms are not alternating between primal and dual), and the fact that the four-arm exponent is strictly smaller than 2. These results were previously known only for Bernoulli percolation ($q = 1$) and the FK-Ising model ($q = 2$). Finally, we prove new bounds on the one, two and four arms exponents for $qin[1,2]$. These improve the previously known bounds, even for Bernoulli percolation.
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In this paper we study the existence phase transition of the random fractal ball model and the random fractal box model. We show that both of these are in the empty phase at the critical point of this phase transition.
For $Delta ge 5$ and $q$ large as a function of $Delta$, we give a detailed picture of the phase transition of the random cluster model on random $Delta$-regular graphs. In particular, we determine the limiting distribution of the weights of the ordered and disordered phases at criticality and prove exponential decay of correlations and central limit theorems away from criticality. Our techniques are based on using polymer models and the cluster expansion to control deviations from the ordered and disordered ground states. These techniques also yield efficient approximate counting and sampling algorithms for the Potts and random cluster models on random $Delta$-regular graphs at all temperatures when $q$ is large. This includes the critical temperature at which it is known the Glauber and Swendsen-Wang dynamics for the Potts model mix slowly. We further prove new slow-mixing results for Markov chains, most notably that the Swendsen-Wang dynamics mix exponentially slowly throughout an open interval containing the critical temperature. This was previously only known at the critical temperature. Many of our results apply more generally to $Delta$-regular graphs satisfying a small-set expansion condition.
We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a generic field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.
We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and letting n go to infinity yields a non-trivial sequence of limit metric spaces C = (C_1, C_2, ...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak, Pittel and Wierman by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component.
In [17], the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in [11], we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem that the magnetization scaled by n 3/4 converges to a specific random variable, with n the number of vertices of random regular graphs.
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