Research
Papers
- Stochastic domination for FK percolation and sharp thinning thresholds for the Ising energy field, joint work with Avelio Sepúlveda. Preprint
[ Abstract | PDF | ArXiv ]
In this work we introduce the concepts of weak domination and weak\(^\dagger\) domination (and analogous notions for the FKG property), which are notions of stochastic dominations after thinning (and sprinkling). Thanks to a new stochastic domination for FK percolation, we are able to show that the energy field of the Ising model (i.e. the set of edges +/+ or -/-) is weakly monotone in the coupling constants and we determine quantitatively the parameter of the thinning necessary to get the domination. We also find the amount of thinning necessary to get the weak\(^\dagger\) domination.
In this paper, we construct in a novel and explicit way the magnetization field of the critical planar Ising model introduced a decade ago. The only randomness that we use is the one arising from the set of interfaces of the spin clusters, the \(\mathrm{CLE}_3\). Therefore, as a corollary of our construction this shows that the critical Ising magnetization field is measurable in term of the \(\mathrm{CLE}_3\). Serious talks at seminars, work-groups and conferences
- A new stochastic domination for FK-percolation talk given at the Summer School PISA [ Abstract ]
FK percolation is a model of bond percolation on a (a priori) finite graph, which depends (as well as on the classical edge parameter \(p\) on a global positive parameter \(q\). The model specializes to classical Bernoulli percolation for \(q=1\), and in the regime \(q\rightarrow 0\) one can recover classical models like the uniform spanning tree or the uniform spanning forest. The case \(q=2\) is deeply connected to the Ising model, one of the main reason for which it has been introduced in the first place. Like Bernoulli percolation, for a given \(q\ge1\), the model (stochastically) increases in the parameter \(p\) but the reason for which it's true is more involved. In this talk, I will present a prove a stochastic domination which strictly improves on this, and actually allows to compare distinct FK-percolation for distinct values of \(q\). This is joint work with Avelio Sepúlveda.
- The magnetization field of critical planar Ising can be reconstructed from its (+/-) interfaces talk given at the workshop Les Diablerets. [ Abstract ]
The planar Ising model at criticality is well understood: on the one hand, it has been shown that the interfaces of the +/- clusters admit a scaling limit, a conformally invariant collection of loops called CLE_3, and on the other hand, convergence of renormalized correlations of the primary fields has been proved as well; the latter result paved the way to prove the existence of another scaling limit: a conformally covariant random field known as the magnetization field. However, no link was made between these two random objects that can be thought of as describing a continuous Ising model. In joint work with Christophe Garban and Avelio Sepúlveda, we give another construction of the magnetization field starting from a CLE_3, showing that in the continuous limit, the field is determined by the loops. In this talk, I will try to convey the main difficulties of the problem and the ideas behind the proof.
