# The conformal loop ensemble nesting field

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## Abstract

The conformal loop ensemble \({{\mathrm{CLE}}}_\kappa \) with parameter \(8/3 < \kappa < 8\) is the canonical conformally invariant measure on countably infinite collections of non-crossing loops in a simply connected domain. We show that the number of loops surrounding an \(\varepsilon \)-ball (a random function of \(z\) and \(\varepsilon \)) minus its expectation converges almost surely as \(\varepsilon \rightarrow 0\) to a random conformally invariant limit in the space of distributions, which we call the nesting field. We generalize this result by assigning i.i.d. weights to the loops, and we treat an alternate notion of convergence to the nesting field in the case where the weight distribution has mean zero. We also establish estimates for moments of the number of CLE loops surrounding two given points.

## Keywords

SLE CLE Conformal loop ensemble Gaussian free field## Mathmatics Subject Classifications

Primary 60J67 60F10 Secondary 60D05 37A25## Notes

### Acknowledgments

Both JM and SSW thank the hospitality of the Theory Group at Microsoft Research, where part of the research for this work was completed. JM’s work was partially supported by DMS1204894 and SSW’s work was partially supported by an NSF Graduate Research Fellowship, award No. 1122374.

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