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COMPASP Report Summary

Project ID: 340340
Funded under: FP7-IDEAS-ERC
Country: Switzerland

Mid-Term Report Summary - COMPASP (Complex analysis and statistical physics)

The goal of the COMPASP project is to achieve breakthroughs in a few outstanding questions in 2D statistical physics, using techniques from complex analysis, probability, dynamical systems, geometric measure theory and theoretical physics. The project emphasizes interaction between mathematics and physics, and conformal symmetries emerging in the scaling limits of lattice models of natural phenomena at criticality, such as the Ising model of a ferromagnet.
The project achieved a number of important results, including:
Geometrical description of the Ising model scaling limit:
Smirnov with coauthors has shown that Ising interfaces converge to Schramm’s SLE curves in the strong sense. Furthermore, Kemppainen and Smirnov developed a full geometrical description of the FK-Ising configuration as a tree of branching SLEs.
Construction of the discrete stress-energy tensor in the loop O(n) model :
Chelkak, Glazman and Smirnov in continued studies of discrete complex structures in lattice models, constructing a discrete version of T(z), which a has a purely geometrical meaning, measuring the response of the partition function of the model to the introduction of discrete conical singularities. In the n=1 case convergence to the continuous counterpart was established.
Continuity of Ising model's spontaneous magnetization:
Duminil-Copin with coauthors has shown that the phase transition of the nearest-neighbor ferromagnetic Ising is continuous. This provides one of the first rigorous results on a statistical physics model in three dimensions, making the first step towards a deeper understanding of 3D critical models.

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