Phase transitions in low-dimensional long-range random field Ising models

TL;DR

This work establishes phase transitions for the long-range RFIM in low dimensions: in 1D, for $1<\alpha<3/2$ without requiring large nearest-neighbor coupling, and in 2D, for $2<\alpha\le 3$, including the critical case $\alpha=3$. The authors introduce a novel multiscale Peierls map with a balancing procedure that handles the randomness of the external field, complemented by energy-entropy bounds and coarse-graining techniques. In 1D, a refined entropy bound and a subgaussian control of the field yield a robust Peierls estimate, while in 2D, a contour-based coarse-graining plus improved energy estimates (notably at the critical exponent) extends previous results to the full long-range region. The results provide a rigorous confirmation of phase transitions in these low-dimensional, long-range RFIMs and clarify the role of the critical exponent $\alpha=3$ in two dimensions.

Abstract

We consider the long-range random field Ising model in dimension $d = 1, 2$, whereas the long-range interaction is of the form $J_{xy} = |x-y|^{-α}$ with $1< α< 3/2$ for $d=1$ and with $2 < α\leq 3$ for $d = 2$. Our main results establish phase transitions in these regimes. In one dimension, we employ a Peierls argument with some novel modification, suitable for dealing with the randomness coming from the external field; in two dimensions, our proof follows that of Affonso, Bissacot, and Maia (2023) with some adaptations, but new ideas are required in the critical case of $α=3$.

Figures (7)

• Figure 1: The figure depicts a configuration $\sigma$ where the red cubes denote plus spins and blue cubes denote minus spins. Taking $M_0=1$ and $\delta=\frac{1}{3}$, the interval $\mathrm{I}_3(x)$ is plus favored, since its $2^{3-1} = 4$ closest neighbors in both directions are positive, and the number of minuses in each interval of the form $\mathrm{I}_3(x \pm 16)$ and $\mathrm{I}_3(x\pm 32)$ is smaller than $\frac{2^3}{2^{3\cdot\frac{1}{3}}} = 4$. Moreover, as there are minuses in $\mathrm{I}_3(x)$ and it is plus favored, it is also isolated.
• Figure 2: An illustration of the balancing procedure. As in Figure \ref{['Fig: plus_favored']}, red regions denote plus spins, while blue regions denote minus spins. We consider $M_0=1$ and $\delta = \frac{2}{3}$. Starting from configuration $\sigma^0$, the first isolated interval selected is $\mathrm{I}_3$. As it is plus favored, it is flipped to plus. Next, $\mathrm{I}_3^\prime$ is selected and flipped to minus for an analogous reason. Then, $\mathrm{I}_4$ is selected and flipped to minus. Notice that $\mathrm{I}_4$ is isolated with respect to $\sigma^2$ but not in $\sigma^1$, since $M_4 = 2^{\frac{8}{3}} \approx 6.34$ and all its first $6$ nearest-neighboring intervals are minus dense with respect to $\sigma^2$ but not $\sigma^1$. Notice that, in $\sigma^3$, $\mathrm{I}_5$ is not plus favored, since $M_5 = 2^{\frac{10}{3}}\approx 10.07$, but not all of its 10 closest neighbors are plus dense. The procedure stops at $\sigma^3$, since the only isolated interval ($\mathrm{I}_9$) is plus favored but contains 0.
• Figure 3: The red region denotes the interval $\mathrm{B}_i$; the black line denotes the interval $\mathrm{C}_i$, and is divided into $16$ parts. The black box delimits the interval $\mathrm{I}^\prime$ that covers more than a $\frac{15}{16}$ fraction of $\mathrm{C}_i$, and the gray box represents $\rho_{\frac{3}{2}}(\mathrm{I}^\prime)$. The dotted line splits $\mathrm{I}^\prime$ in half.
• Figure 4: Considering a configuration $\sigma$ in an interval $\mathrm{I}$, and taking $A=\mathrm{I}^-(\sigma)$, the picture depicts, from bottom to top, $\Psi_0(A), \Psi_1(A), \Psi_2(A)$ and $\Psi_3(A)$. For every $\ell=0,1,2,3$, intervals $\mathrm{I}_\ell$ are painted red if $\Psi_\ell(A, \mathrm{I}_\ell) = 1$, painted blue if $\Psi_\ell(A, \mathrm{I}_\ell) = -1$ and painted white otherwise. We write $\sigma = \Psi_0(A)$ since both functions attribute the same value to each site.
• Figure 5: The gray region delimits the interval $\mathrm{I}$. We take $\ell$ satisfying $\frac{15}{8}2^{\ell-2}\leq |\mathrm{I}| \leq \frac{15}{8}2^{\ell-1}$. The black line represents $\mathbb{Z}$ divided into intervals of length $2^{\ell}/16$. The $\ell$-interval $\mathrm{I}_{\ell}$ with endpoints in the red regions satisfies the desired inequality.

Theorems & Definitions (78)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof