The random field Ising chain domain-wall structure in the large interaction limit

TL;DR

This work rigorously justifies the disorder-driven domain-wall picture in the random-field Ising chain with nearest-neighbor interactions in the $J\to\infty$ regime. It constructs a transfer-matrix framework to analyze the infinite-volume Gibbs measure, introduces the Fisher domain-wall configuration via Γ-extrema of a two-sided random walk, and proves that typical spin configurations converge, in a precise density sense, to a disorder-dependent Fisher pattern with discrepancy density $D_\Gamma=O(\log_{\circ 2}(\Gamma)/\Gamma)$. The authors connect the discrete model to Fisher renormalization group predictions by linking the spin configuration to a pair of hard-wall Markov chains and to a Neveu–Pitman-type process in the continuum limit. The results highlight a universal mechanism by which strong disorder reshapes ground-state-like structures in one-dimensional random-field systems and provide a quantitative bridge between discrete and continuum RG descriptions.

Abstract

We study the configurations of the nearest neighbor Ising ferromagnetic chain with IID centered and square integrable external random field in the limit in which the pairwise interaction tends to infinity. The available free energy estimates for this model show a strong form of disorder relevance, i.e., a strong effect of disorder on the free energy behavior, and our aim is to make explicit how the disorder affects the spin configurations. We give a quantitative estimate that shows that the infinite volume spin configurations are close to one explicit disorder dependent configuration when the interaction is large. Our results confirm predictions on this model obtained in D. S. Fisher, P. Le Doussal and C. Monthus (Phys. Rev. E 2001) by applying the renormalization group method introduced by D. S. Fisher (Phys. Rev. B 1995).

Figures (7)

• Figure 1: The sequence of $\Gamma$-extrema for the simple symmetric random walk with $\Gamma=5/2$: in fact, $\Gamma\in (2,3]$ leads to the very same sequence of $\Gamma$ extrema.
• Figure 2: The function $b_\Gamma$ for $\Gamma=4$ (on the left) and $\Gamma=10$ (on the right). The dashed lines plot the function $\widehat{b}_\Gamma$ introduced in § \ref{['sec:constrained']}.
• Figure 3: A simulation of the $(l_n)_{n=0,\ldots,N}$, with $l_0=0$ and $N= 5\cdot 10^7$, $\Gamma=20$. In the first case the law of $h_1$ is $(\delta_{-2}+\delta_{+2})/2$ and in the second case the law is $(\delta_{-2}+ \gamma)/2$ where $\gamma$ is the law of a Gaussian of mean $2$ and variance $1/ \sqrt{2}$. These two histograms are expected to give an idea of how the invariant probability looks like: in fact, the simulation is very stable with respect to increasing $N$ and changing the randomness. Note that this empirical observation is compatible in the second case with the invariant probability being very close to (a multiple) of the Lebesgue measure away from the boundaries (this result is proven in cf:GG22 under stronger regularity properties of the law of $h_1$). However, in the first case a periodic phenomenon seems to be present and proximity to the Lebesgue measure appears to be plausible only if averages over large boxes are taken.
• Figure 4: In this figure $\Gamma=5.5$. The $\widehat{l}$ process, on the right, is driven by twice the random walk trajectory, on the left. We look at the trajectory from $\mathtt{s}^{\downarrow}(\Gamma)$ and we use $\widehat{l}_{\mathtt{s}^{\downarrow}(\Gamma)}\in [-\Gamma, \Gamma]$. The fact that the $S$ trajectory increases of at least $\Gamma$ from $\mathtt{s}^{\downarrow}(\Gamma)$ to $\mathtt{v}^{\downarrow}(\Gamma)$ guarantees that the $\widehat{l}$ trajectory hits $\Gamma$ not later than $\mathtt{v}^{\downarrow}(\Gamma)$. Moreover, $\widehat{l}_{\mathtt{v}^{\downarrow}(\Gamma)}=\Gamma$ too. After $\mathtt{v}^{\downarrow}(\Gamma)$ and up to time 0 the $\widehat{l}$ trajectory copies the increments of $2S$.
• Figure 5: In this figure $\Gamma=5.5$. Equation \ref{['eq:explicit-hat-r']} holds because the evolution of $(\widehat{r}_{-n})$, i.e. we are looking from right to left, repeats the increments of $(-2S_{-n-1})$ except that the $\widehat{r}$ process is confined to $[-\Gamma, \Gamma]$. Always arguing with the reversed time arrow, we see that, regardless of the value of $\widehat{r}_{\mathtt{t}^\downarrow (\Gamma) +1 }\in [-\Gamma, \Gamma]$, the $\widehat{r}$ process is going to hit $\Gamma$ not later than $\mathtt{u}^\downarrow (\Gamma)$ and that, in any case, $\widehat{r}_{\mathtt{u}^\downarrow (\Gamma)+1}= -\Gamma$. After that moment and up to time 1 the evolution of $\widehat{r}$ reproduces the increments of $(-2S_{-n-1})$.

Theorems & Definitions (47)

• Remark 1.1
• Theorem 1.2
• Proposition 1.3
• Remark 1.4
• Remark 2.1
• Lemma 2.2
• Remark 2.3
• proof
• Proposition 2.4
• Proposition 2.5