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## Forthcoming talks:

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Thursday 721.11.2019 C124 14-16 o'clock

Eero Saksman:

**SEMINAARI PERUTTU TÄLTÄ PÄIVÄLTÄ SEMINAR TODY CANCELLED!**

I'm sorry, but forgot to cancel the seminar from the web-page! Eero. S.

Thursday 14.11.2019 C124 14-16 o'clock clock (joint talk with Mathematical physics seminar)

Mikhail Basok Anton Nazarov (Saint Petersburg State University)

**Tau-functions à la Dubédat and cylindrical events in the double-dimer model **

Double-dimer model on a given graph is obtained by sampling two independent dimer configurations taken uniformly at random: this produces a number of loops and double edges, removing the latter one obtains what is called double dimer loop ensemble. Given a simply-connected domain on the complex plane consider its approximation by a sequence of Temperley domains on a square grid, where the step of the grid tends to zero. It was predicted by Kenyon that the corresponding sequence of double dimer loop ensembles converges to Conformal Loop Ensemble with parameter 4 (CLE(4)) sampled in the original domain. Recently this conjecture was supported by a breakthrough work of Dubedat: in this work a large family of observables, called topological correlators, was constructed and it was shown that their values for double-dimer loop ensembles converges to their values for CLE(4). Topological correlators carry a lot of topological information about loops by their construction and it was reasonable to expect that their values determine the probability measure on families of loops uniquely. As it turned out, values of topological correlators determine probabilities of a large class of events called cylindrical, which, in particular, implies the claim above: topological correlators do characterize a measure. In this talk we will discuss the construction of topological correlators and the machinery developed to extract concrete probabilities from their values. Based on a joint work with Dmitry Chelkak (Paris).

Thursday 21

**Limit shape for the $so(2n+1)$ Lie algebras in the infinite rank limit and an electrostatic problem. **

Abstract:

We consider a tensor power of the spinor representation of the Lie algebra $so(2n+1)$. Tensor product decomposition into irreducible representations leads to the appearance of probability measure on the set of dominant integral weights. We consider the behavior of this measure in the limit of infinite tensor power and infinite rank of the algebra.

We show that in this limit the measure is concentrated in the single weight, thus the limit shape phenomenon is observed. The coordinates of this weight can be seen as the positions of charged particles on the line. This particles repulse each other but are confined in a closed interval by an external potential. We show that the limit shape is described by the density function of these particles.

Charge density function is the solution of a variational problem for the equilibrium measure, thus our result is comparable to various results in theory of random matrices and orthogonal polynomials.

Our result is similar to famous Vershik-Kerov-Logan-Shepp limit shape for the Plancherel measure on Young diagrams, since the limit shape for Young diagrams can be obtained from tensor product decomposition of $sl(n)$-representations.

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## Previous talks:

Thursday 14.11.2019 C124 14-16 o'clock clock (joint talk with Mathematical physics seminar)

Anton Nazarov Mikhail Basok (Saint Petersburg State University)

**Limit shape for the $so(2n+1)$ Lie algebras in the infinite rank limit and an electrostatic problem. **

Abstract:

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**Tau-functions à la Dubédat and cylindrical events in the double-dimer model **

Double-dimer model on a given graph is obtained by sampling two independent dimer configurations taken uniformly at random: this produces a number of loops and double edges, removing the latter one obtains what is called double dimer loop ensemble. Given a simply-connected domain on the complex plane consider its approximation by a sequence of Temperley domains on a square grid, where the step of the grid tends to zero. It was predicted by Kenyon that the corresponding sequence of double dimer loop ensembles converges to Conformal Loop Ensemble with parameter 4 (CLE(4)) sampled in the original domain. Recently this conjecture was supported by a breakthrough work of Dubedat: in this work a large family of observables, called topological correlators, was constructed and it was shown that their values for double-dimer loop ensembles converges to their values for CLE(4). Topological correlators carry a lot of topological information about loops by their construction and it was reasonable to expect that their values determine the probability measure on families of loops uniquely. As it turned out, values of topological correlators determine probabilities of a large class of events called cylindrical, which, in particular, implies the claim above: topological correlators do characterize a measure. In this talk we will discuss the construction of topological correlators and the machinery developed to extract concrete probabilities from their values. Based on a joint work with Dmitry Chelkak (Paris).

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Thursday 10.10.2019 C124 14-16 o'clock

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