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2 results

  • 8 - SLE and the Quantum Zipper

  • Nathanaël Berestycki, Universität Wien, Austria, Ellen Powell, Durham University
  • Book:
    Gaussian Free Field and Liouville Quantum Gravity
    Published online:
    20 November 2025
    Print publication:
    04 December 2025, pp 299-333

  • Summary

    We describe couplings between Schramm–Loewner Evolution (SLE) curves and variants of the Gaussian free field (GFF). In particular, we give a complete proof of Sheffield’s construction of γ-quantum boundary length along an SLEγ2 curve, as measured by an independent underlying GFF. The main input for this proof is a rigorous construction of the so-called quantum gravity zipper, which is a stationary dynamic on quantum surfaces (defined using a GFF) decorated by SLE. Another consequence of this construction is that drawing an SLE curve on top of an appropriate independent quantum surface splits the surface into two independent and identically distributed (sub)-surfaces, glued according to boundary length. In particular, this shows that SLE curves are solutions of natural random conformal welding problems.

Gaussian Free Field and Liouville Quantum Gravity
  • Nathanaël Berestycki, Ellen Powell
  • Published online:
    20 November 2025
    Print publication:
    04 December 2025
  • In this comprehensive volume, the authors introduce some of the most important recent developments at the intersection of probability theory and mathematical physics, including the Gaussian free field, Gaussian multiplicative chaos and Liouville quantum gravity. This is the first book to present these topics using a unified approach and language, drawing on a large array of multi-disciplinary techniques. These range from the combinatorial (discrete Gaussian free field, random planar maps) to the geometric (culminating in the path integral formulation of Liouville conformal field theory on the Riemann sphere) via the complex analytic (based on the couplings between Schramm–Loewner evolution and the Gaussian free field). The arguments (currently scattered over a vast literature) have been streamlined and the exposition very carefully thought out to present the theory as much as possible in a reader-friendly, pedagogical yet rigorous way, suitable for graduate students as well as researchers.