Smirnov's convergence conjecture for the Potts parafermionic observable

Fix q4q\le 4 and p=pcp=p_c. Let (Ωδ,aδ,bδ)(\Omega_\delta,a_\delta,b_\delta) be Dobrushin domains approximating a simply connected domain Ω{\bf \Omega} with marked boundary points a{\bf a} and b{\bf b}, and let fδf_\delta be the vertex parafermionic observable obtained by averaging the edge fermionic observable over neighboring edges. Let σ\sigma be the parameter used in the source, and let ϕ\phi be a conformal map from Ω{\bf \Omega} to the strip R×(0,1)\mathbb{R}\times(0,1) mapping a{\bf a} to -\infty and b{\bf b} to \infty. Smirnov's observable convergence conjecture.

limδ0(2δ)σfδ=(ϕ)σ.\lim_{\delta\to0}(2\delta)^{-\sigma}f_\delta=(\phi')^\sigma.

The source explains that this convergence would imply the preceding conformal-invariance conjecture, but does not state that it has been proved in the full range q4q\le4.

Sources & referencesView supporting material

Primary source

Hugo Duminil-Copin, “Lectures on the Ising and Potts models on the hypercubic lattice”, arXiv:1707.00520 (2017).

Additional references

3 papers in this index state this conjecture (2012–2017). The statement above is taken from the most recent of them; the others are arXiv:1505.04159, arXiv:1208.3787.

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