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Six-vertex model and Schramm-Loewner evolution.

Richard Kenyon1, Jason Miller2, Scott Sheffield3

  • 1Department of Mathematics, Brown University, 1 Prospect Street, Providence, Rhode Island 02912, USA.

Physical Review. E
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Summary
This summary is machine-generated.

We linked square ice configurations to space-filling Peano curves. Their scaling limit is a fractal curve, Schramm-Loewner evolution (SLE), with unusual parameters outside the classical range.

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Area of Science:

  • Statistical Mechanics
  • 2D Ice Models
  • Conformal Field Theory

Background:

  • Square ice is a statistical mechanics model for 2D ice.
  • It is widely believed to possess a conformally invariant scaling limit.

Purpose of the Study:

  • To associate a Peano (space-filling) curve to square ice and six-vertex model configurations.
  • To determine the scaling limit of these configurations and identify the corresponding Schramm-Loewner evolution (SLE) parameter κ.

Main Methods:

  • Associating Peano curves to square ice and six-vertex model configurations.
  • Analyzing the scaling limit of these associated curves using fractal geometry concepts.

Main Results:

  • The scaling limit of square ice configurations is identified as a space-filling version of SLE with κ=12.
  • For the six-vertex model at its free-fermion point, the scaling limit corresponds to SLE with κ=8+4√3.
  • These κ values (12 and 8+4√3) fall outside the classical interval of 2≤κ≤8.

Conclusions:

  • The study establishes a connection between statistical mechanics models of ice and fractal random curves.
  • It reveals novel, non-classical values for the Schramm-Loewner evolution parameter κ in these systems.