Smirnov's convergence conjecture for the Potts parafermionic observable

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Fix q≤4q\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.

lim⁡δ→0(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 q≤4q\le4.

References

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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