Dynamical random field Ising model at zero temperature

TL;DR

The paper studies the zero-temperature evolution of the RFIM as the external-field mean M increases, comparing ground-state evolution with zero-temperature Glauber dynamics. It establishes a sharp dimension-dependent picture: in 2D there is no global avalanche for any disorder strength, while in d≥3 a disorder-driven phase transition governs the occurrence of global avalanches, with small ε yielding a macroscopic flip and large ε suppressing it. A central methodological contribution is a multi-scale, isoperimetric approach that connects RFIM dynamics to polluted bootstrap percolation, enabling precise probabilistic control of flip sets and triggering times. The work also develops a rigorous polluted bootstrap percolation framework and leverages it to derive threshold results and to solve an open problem in Gravner-Holroyd-3D, with extensions to Glauber dynamics at positive temperature and open questions for higher dimensions. Overall, the paper provides a cohesive, dimension-aware treatment of dynamical RFIM behavior at zero temperature and establishes deep connections to percolation theory and multi-scale energy-change control.

Abstract

In this paper, we study the evolution of the zero-temperature random field Ising model as the mean of the external field $M$ increases from $-\infty$ to $\infty$. We focus on two types of evolutions: the ground state evolution and the Glauber evolution. For the ground state evolution, we investigate the occurrence of global avalanche, a moment where a large fraction of spins flip simultaneously from minus to plus. In two dimensions, no global avalanche occurs, while in three or higher dimensions, there is a phase transition: a global avalanche happens when the noise intensity is small, but not when it is large. Additionally, we study the zero-temperature Glauber evolution, where spins are updated locally to minimize the Hamiltonian. Our results show that for small noise intensity, in dimensions $d =2$ or $3$, most spins flip around a critical time $c_d = \frac{2 \sqrt{d}}{1 + \sqrt{d}}$ (but we cannot decide whether such flipping occurs simultaneously or not). We also connect this process to polluted bootstrap percolation and solve an open problem on it.

Figures (11)

• Figure 1: An illustration that ${\rm Flip}(M)$ does not intersect any $L$-good box. The subset ${\rm Flip}(M)$ is colored in blue, the boundary of $B_{2L}(u)$ is colored in green and dashed green, and the boundary of $B_{L}(u)$ is colored in red. The change of boundary condition of $B_{2L}(u)$ (the dashed green) affects the configuration in $B_L(u)$, and thus $B_L(u)$ is not $L$-good.
• Figure 2: An illustration for $\mathbf{C}^{j}$. Plus spins are colored in red and minus spins are colored in blue. The set $\mathbf{C}^1$ is enclosed by the dashed black contour, the set $\mathbf{C}^2$ is enclosed by the dashed orange contour, and the sets $\mathbf{C}^3$ is the part outside the dashed green contour in the figure. In this figure, we have $\mathbf{C}^1 = \mathsf{C}^1$.
• Figure 3: A spin configuration with a plus global spin cluster. Here we color the plus spins in blue and minus spins in lightblue. Note that there exists a blue cluster such that all the other connected components after removing this cluster have $|\cdot|_{\infty}$-diameters at most $\widetilde{N}$.
• Figure 4: An illustration of case \ref{['def:good-box-case-b']} for a good $L_n$-box $B(x, L_n)$. The boundary of $B(y, \sqrt{K}L_{n-1})$ is colored in green, and the boundary of $B(y, 3L_{n-1})$ is colored in blue. The bad $L_{n-1}$-boxes, colored in red, are all contained in $B(y, 3L_{n-1})$. Note that we allow bad $L_{n-2}$-boxes to exist in $B(x, L_n)\setminus B(y, 3L_{n-1})$, but they are not illustrated. In the final configuration of $B(x,L_n)$, all the open clusters that intersect $B(y, 3L_{n-1})$ are contained in $B(y, \sqrt{K}L_{n-1})$.
• Figure 5: An illustration for $d=2$ and $\boldsymbol{k}=(1,1)$. For the (nice) marked box $(B(x,L),y(x))$, the chosen vertex $y(x)$ is colored in red. The oriented shield ${\rm Sh}(x,\boldsymbol{k})$ is also colored in red with $y(x)$ as the corner. The domain $B^+_{\boldsymbol{k}}(x,5^{10d}L)$ is colored in green, and the domain $B^-_{\boldsymbol{k}}(x,5^{10d}L)$ is colored in light purple. The boundary of $\mathsf{R}(y(x))$ is shown with dashed black lines. Although in the two-dimensional case the set $\mathsf{R}(y(x))$ contains ${\rm Sh}(x,\boldsymbol{k})$, this is not true in three dimensions or higher.

Theorems & Definitions (73)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6: PollutedBP_originalGravner-Holroyd-3D
• Remark 1.7: Short-range interactions
• Lemma 2.1
• proof
• Lemma 2.2