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The dynamical Ising-Kac model in 3D converges to .

P Grazieschi1, K Matetski2, H Weber3

  • 1University of Bath, Bath, UK.

Probability Theory and Related Fields
|February 27, 2025
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Summary

This study analyzes spin dynamics in a 3D Ising-Kac model, showing its fluctuations converge to a nonlinear stochastic partial differential equation (SPDE) near the critical temperature, confirming a long-standing conjecture.

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Area of Science:

  • Statistical Mechanics
  • Mathematical Physics
  • Dynamical Systems

Background:

  • The Glauber dynamics of the ferromagnetic Ising-Kac model are investigated on a 3D lattice.
  • The spin flipping rate depends on a large-neighborhood average field.
  • Previous work explored the 2D case and posed conjectures for 3D.

Purpose of the Study:

  • To analyze the random fluctuations of a coarse-grained spin field in the 3D Ising-Kac model.
  • To demonstrate convergence to a specific nonlinear stochastic partial differential equation (SPDE).
  • To rigorously settle a conjecture regarding this convergence.

Main Methods:

  • Analysis of Glauber dynamics on a 3D periodic lattice.
  • Study of rescaled coarse-grained spin field fluctuations in the thermodynamic and large neighborhood limits.
  • Application of the regularity structures framework for solving the nonlinear SPDE.

Main Results:

  • The process converges in distribution to the solution of the dynamical \phi^4 model on a torus near the mean-field critical temperature.
  • This convergence confirms a conjecture by Giacomin et al. (1999).
  • The renormalization of the SPDE corresponds to a small shift in the inverse temperature.

Conclusions:

  • The study provides a rigorous mathematical framework for understanding the macroscopic behavior of the 3D Ising-Kac model near criticality.
  • It establishes a connection between discrete spin dynamics and continuous SPDEs.
  • The findings highlight the importance of renormalization in bridging microscopic and macroscopic descriptions in statistical physics.