Smirnov's O(n)-model interface conjecture

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Fix n∈[0,2]n\in[0,2] and consider the loop O(n)O(n) model with an interface between boundary points aa and bb. Smirnov's O(n)-model interface conjecture. For z=1/2+2−nz=1/\sqrt{2+\sqrt{2-n}}, the interface converges, as the lattice spacing tends to zero, to

SLEκwithκ=4π2π−arccos⁡(−n/2).\mathrm{SLE}_\kappa\quad\text{with}\quad\kappa=\frac{4\pi}{2\pi-\arccos(-n/2)}.

For z>1/2+2−nz>1/\sqrt{2+\sqrt{2-n}}, it converges to

SLEκwithκ=4πarccos⁡(−n/2).\mathrm{SLE}_\kappa\quad\text{with}\quad\kappa=\frac{4\pi}{\arccos(-n/2)}.

This extends the proved n=1n=1, z=1/3z=1/\sqrt{3} case, where the interface converges to SLE3\mathrm{SLE}_3; the other parameter regimes are presented as conjectural.

References

Primary source

Roland Bauerschmidt, Hugo Duminil-Copin, Jesse Goodman and Gordon Slade, “Lectures on Self-Avoiding Walks”, arXiv:1206.2092 (2012).

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