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Near-critical scaling limits

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Consider percolation on the triangular grid in the plane. The scaling limit (as n goes to infinity) of a critical rescaled percolation \omega_{p_c}n on the triangular grid of mesh 1/n is a very rich probabilistic object (it is the analog of the Brownian motion for the Random walk), and is now well understood thanks to the works of Smirnov and Schramm.

The purpose of a joint work with Gabor Pete and Oded Schramm is to ``visualize’’ the phase transition which occurs at p_c, from the perspective of the scaling limit. A first natural attempt in this direction is to consider the scaling limit as n goes to infinity of non-critical rescaled percolations \omega_pn (with p \neq p_c). Unfortunately, such scaling limits are “degenerate”. In order to obtain non-trivial “off-critical” scaling limits, $p$ and $n$ need to be rescaled accordingly.

I will describe in this talk what the natural renormalization is if one wishes to observe the emergence of an infinite cluster, seen from the continuous. The main result in joint work with G. Pete and O. Schramm is that “up to scaling”, there is a unique near-critical scaling limit. This near-critical limit is not conformally invariant anymore, but one can nevertheless give a precise description of its “conformal defect”.

Finally, I will discuss some new results about dynamical and near-critical regimes in the case of dependent models like the Ising model or the Random-Cluster model.

This talk is part of the Probability series.

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