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Complex analysis and statistical physics

Final Report Summary - COMPASP (Complex analysis and statistical physics)

Over the last two decades there were several breakthroughs in our understanding of lattice models of 2D statistical physics and their scaling limits, such as the Ising model of a ferromagnet, the percolation model of a liquid seeping through a porous medium and so on. The goal of the COMPASP project was to build upon those and combine tools from probability, combinatorics and complex analysis to achieve progress in several difficult and important open questions.
The project emphasized interaction between mathematics and physics, and also led to interdisciplinary research between mathematics and biology.

The project led to 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, Antti Kemppainen and Smirnov gave a full geometrical description of the FK-Ising configuration as a tree of branching SLEs, and developed machinery for similar descriptions for other models.
- Conformal invariance of percolation:
Khristoforov (PhD student) and Smirnov obtained a new proof of the conformal invariance of the percolation on triangular lattice, viewed as O(1) loop model on the hexagonal. Besides being more conceptual, it leads to new results, such as the derivation of the probability of an interface passing above a point.
- Universality in 2D lattice models:
Russkikh (PhD student) in two works (joint with R. Kenyon, W. Lam, S. Ramassamy and with D. Chelkak, B. Laslier) obtained universality results for dimer models. The new family of "origami" graphs was introduced, and seems to be the most general and appropriate setting for planar universality results.
- Construction of the discrete stress-energy tensor in the loop O(n) model:
Chelkak, Glazman and Smirnov constructed 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 (who participated in the project before getting his own ERC grant) 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.
- Ising model and cellular automata:
Smirnov with biology group of Michel Milinkovitch (UniGe 0e partment of Genetics and Evolution) showed that skin coloring in a certain species of lizards is generated by a Turing’s reaction-diffusion equation, leading to cellular automata, closely related to the planar Ising model. The research was published in Nature and got a lot of attention.