The Heat Equation Shrinks Ising Droplets to Points

Lacoin, Hubert; Simenhaus, François; Toninelli, Fabio Lucio

doi:10.1002/cpa.21533

Cited by 6 publications

(11 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This belief is supported by numerical simulations (for this and related models, see [5] and references in [25,10]), heuristic arguments [10] and partial mathematical results [25,1]. Let us also mention that, for the zero-temperature two-dimensional (and not three-dimensional) Ising model, convergence of the evolution of spin droplets to a deterministic equation of anisotropic mean-curvature type under diffusive scaling has been achieved very recently [15,16]. The limit equation is somewhat the analog of (1), with the notable difference that in that case φ describes a curve in the plane and not a surface in threedimensional space.…”

Section: Introductionmentioning

confidence: 76%

Lozenge Tilings, Glauber Dynamics and Macroscopic Shape

Laslier¹,

Toninelli

2015

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract. We study the Glauber dynamics on the set of tilings of a finite domain of the plane with lozenges of side 1 L. Under the invariant measure of the process (the uniform measure over all tilings), it is well known [4] that the random height function associated to the tiling converges in probability, in the scaling limit L → ∞, to a non-trivial macroscopic shape minimizing a certain surface tension functional. According to the boundary conditions the macroscopic shape can be either analytic or contain "frozen regions" (Arctic Circle phenomenon [3,11]).It is widely conjectured, on the basis of theoretical considerations [23,10], partial mathematical results [25,1] and numerical simulations for similar models ([5], cf. also the bibliography in [25,10]), that the Glauber dynamics approaches the equilibrium macroscopic shape in a time of order L 2+o(1) . In this work we prove this conjecture, under the assumption that the macroscopic equilibrium shape contains no "frozen region".

show abstract

Section: Introductionmentioning

confidence: 76%

Lozenge Tilings, Glauber Dynamics and Macroscopic Shape

Laslier¹,

Toninelli

2015

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…In the intermediate regime when the droplet is still very large compared to the lattice spacing, but already very small compared to the initial size, the droplet has an essentially deterministic shape (since fluctuations are negligible in comparison with the characteristic size of the droplet) and the initial condition no longer matters. In the case of the classical Ising model, this limiting shape has been analytically determined [11]; see also [21] and the proof of the convergence to the limiting shape [22]. It appears possible to generalize these results to include long-ranged interactions.…”

Section: Discussionmentioning

confidence: 90%

Limiting shapes in two-dimensional Ising ferromagnets

Krapivsky

Olejarz

2013

Phys. Rev. E

View full text Add to dashboard Cite

We consider an Ising model on a square lattice with ferromagnetic spin-spin interactions spanning beyond nearest neighbors. Starting from initial states with a single unbounded interface separating ordered phases, we investigate the evolution of the interface subject to zero-temperature spin-flip dynamics. We consider an interface which is initially (i) the boundary of the quadrant or (ii) the boundary of a semi-infinite bar. In the former case the interface recedes from its original location in a self-similar diffusive manner. After a rescaling by √[t], the shape of the interface becomes more and more deterministic; we determine this limiting shape analytically and verify our predictions numerically. The semi-infinite bar acquires a stationary shape resembling a finger, and this finger translates along its axis. We compute the limiting shape and the velocity of the Ising finger.

show abstract

“…A set with boundary following this equation is known to shrink to a point in finite time for a wide range of anisotropies a, see e.g. [LST14a] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…In two dimensions, a landmark is the proof of anisotropic motion by curvature for the zero temperature Ising model with Glauber dynamics (or zero-temperature stochastic Ising model). The drift at time 0 was computed in [CL07] before the full motion by curvature (1.2) was proven in [LST14b]- [LST14a]. Their proof crucially relies on monotonicity of the Glauber dynamics.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Motion by curvature and large deviations for an interface dynamics on Z 2

Dagallier

2020

Preprint

View full text Add to dashboard Cite

We study large deviations for a Markov process on curves in Z 2 mimicking the motion of an interface. Our dynamics can be tuned with a parameter β, which plays the role of an inverse temperature, and coincides at β = ∞ with the zero-temperature Ising model with Glauber dynamics, where curves correspond to the boundaries of droplets of one phase immersed in a sea of the other one. We prove that contours typically follow a motion by curvature with an influence of the parameter β, and establish large deviations bounds at all large enough β < ∞. The diffusion coefficient and mobility of the model are identified and correspond to those predicted in the literature.

show abstract

The Heat Equation Shrinks Ising Droplets to Points

Cited by 6 publications

References 27 publications

Lozenge Tilings, Glauber Dynamics and Macroscopic Shape

Lozenge Tilings, Glauber Dynamics and Macroscopic Shape

Limiting shapes in two-dimensional Ising ferromagnets

Motion by curvature and large deviations for an interface dynamics on Z 2

Contact Info

Product

Resources

About