# Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)

### Mikhail Basok

St. Petersburg State University and Steklov Mathematical Institute, St. Petersburg, Russia### Dmitry Chelkak

Ecole Normale Supérieure, Paris, France and Steklov Mathematical Institute, St Petersburg, Russia

## Abstract

Building upon recent results of Dubédat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations $Ω_{δ}$ to a simply connected domain $Ω⊂C$ we prove the convergence of probabilities of cylindrical events for the *double-dimer loop ensembles* on $Ω_{δ}$ as $δ→0$. More precisely, let $λ_{1},…,λ_{n}∈Ω$ and $L$ be a macroscopic lamination on $Ω∖{λ_{1},…,λ_{n}}$, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities $P_{L}$ that one obtains $L$ after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on $Ω_{δ}$ converge to a conformally invariant limit $P_{L}$ as $δ→0$, for each $L$.

Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety Hom$(π_{1}(Ω∖{λ_{1},…,λ_{n}})→SL_{2}(C)$ and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do *not* use any RSW-type arguments for double-dimers.

The limits $P_{L}$ of the probabilities $P_{L}$ are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock–Goncharov lamination basis on the representation variety. The fact that $P_{L}$ coincides with the probability of obtaining $L$ from a sample of the nested CLE(4) in $Ω$ requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.

## Cite this article

Mikhail Basok, Dmitry Chelkak, Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4). J. Eur. Math. Soc. 23 (2021), no. 8, pp. 2787–2832

DOI 10.4171/JEMS/1072