Validity and explanatory power of the finite-variable reduced dynamics for Glauber dynamics of the bond-diluted Ising model on a Bethe lattice

Ohta (2010) studied Glauber single-spin-flip dynamics of a bond-diluted ferromagnetic Ising model on a Bethe lattice and derived a finite-variable reduced dynamics — a closed set of rate equations for the population weights of a reference spin's local environments — which agrees with Monte Carlo simulations in almost all situations, and which itself exhibits a multi-step relaxation of the magnetization in the low-temperature part of the Griffiths phase without bond-percolation clusters, but rests on unproven assumptions. The paper leaves five linked problems open: (i) prove or disprove the comparison conjecture $\tau_{\mathrm{d}}\le\tau$ and $m_{\mathrm{d}}(t)\le m(t)$ for $m(0)=1$ between the reduced and true dynamics; (ii) construct a quantitative theory of the threshold temperature $T_{\mathrm{th}}(p,c)$ at which multi-step relaxation onsets, connecting the reordering of equilibrium local-population weights to distinct observable timescales; (iii) extend the finite-variable closure so it gradually recovers the nonexponential long-time Griffiths-phase relaxation for $p>p_c$; (iv) clarify the relation between this approximation and dynamical replica theory for quenched-disordered Bethe systems; (v) assess the approximation's validity under replica-symmetry breaking. No resolution of the five items is known in the literature to date.

Background

A Bethe lattice is an infinite loopless graph in which every site has a fixed coordination number $c$ (so it is locally tree-like and is the mean-field reference geometry for sparse systems). Consider a ferromagnetic Ising model on such a graph in which each bond is independently removed with probability $1-p$ (bond dilution). For $p$ above the bond-percolation threshold $p_c=1/(c-1)$, an infinite cluster of connected bonds exists; for $p<p_c$ all connected clusters are finite. In both cases the model has a Griffiths phase, a non-ordered regime in which rare large connected clusters are locally ordered, relax on exponentially widely separated timescales, and produce anomalous nonexponential slow relaxation. Glauber dynamics is the standard single-spin-flip Markov process that equilibrates the Ising model at temperature $T$ with each flip rate depending on the local field of its neighbors. Ohta (2010, J. Phys. A 43, 395003) simulated this dynamics by Monte Carlo and approximated it by a reduced dynamics: instead of tracking all spins, one tracks the time evolution of $\rho_{\sigma,l,u}$, the probability that a reference site has spin value $\sigma$, $l$ intact bonds to its neighbors (its bonded sites), and $u$ downward spins (spin value $-1$) among those bonded sites. The equations for these probabilities close only after one approximation: the conditional distribution of a neighbor's environment $(l,u)$ given a bonded neighbor's full state is replaced by the conditional distribution given only the neighbor's spin. This yields a finite closed set of ordinary differential equations (a finite-variable approximation, reducing at $p=1$ to the independent-neighbor approximation of Semerjian and Weigt 2004), which is exact for equilibrium properties and whose solutions agree with the Monte Carlo simulations in almost all situations. In the low-temperature part of the Griffiths phase without bond-percolation clusters ($p<p_c$) the reduced dynamics exhibits a multi-step relaxation — the magnetization $m(t)$ per site decaying through a sequence of steps, operationally multiple inflection points of $\log m$ versus $\log t$ — appearing below a threshold temperature $T_{\mathrm{th}}(p,c)$ (for example $T_{\mathrm{th}}\simeq 0.6$ at $c=3$, $p=1/5$), and the Monte Carlo simulations confirm the reduced dynamics where they can resolve it. Ohta conjectured, on numerical evidence, the comparison inequalities $\tau_{\mathrm{d}}\le\tau$ (reduced relaxation time bounds the true one from below) and $m_{\mathrm{d}}(t)\le m(t)$ for quenches from $m(0)=1$, where $\tau$ is the characteristic relaxation time of $m(t)$ and the subscript d denotes the same quantities computed from the reduced dynamics. However, the approximation neglects correlations beyond the reference environment, its accuracy is uncontrolled, it fails to capture the nonexponential long-time tails in the Griffiths phase with percolating clusters ($p>p_c$), and its relation to other systematic closures — dynamical replica theory (DRT, a technique that analyzes the dynamics of disordered systems by replicating the quenched disorder and assuming replica symmetry, applied by Hatchett et al. 2005 and Mozeika and Coolen 2009 to finitely connected diluted Ising systems) and cavity master equations (dynamical message-passing closures developed by Aurell and collaborators) — was left unclarified. Replica-symmetry breaking (RSB) denotes the failure of the assumption that the replicated disorder average is described by a single saddle point; it signals many pure thermodynamic states and invalidates single-state dynamical closures on tree-like graphs.

Problem Statement

Establish the validity domain and explanatory power of Ohta's finite-variable reduced dynamics for the Glauber dynamics of the bond-diluted ferromagnetic Ising model on a Bethe lattice of coordination number $c$ at bond-retention probability $p$ and temperature $T$. This single objective comprises five linked open questions, all stated explicitly in the source paper: (i) prove or disprove the comparison conjecture that the reduced dynamics relaxes at least as fast as the true dynamics, $\tau_{\mathrm{d}}\le\tau$ and $m_{\mathrm{d}}(t)\le m(t)$ for quenches from full magnetization $m(0)=1$; (ii) explain the mechanism of the threshold temperature $T_{\mathrm{th}}(p,c)$, below which multi-step relaxation appears in the Griffiths phase without percolating clusters (e.g. $T_{\mathrm{th}}\simeq 0.6$ at $c=3$, $p=1/5$) — construct a quantitative theory linking the temperature-dependent reordering of the equilibrium local-population weights $\rho_{+1,l,u}^{\mathrm{eq}}$ to the emergence of the distinct observable timescales of multi-step relaxation; (iii) extend the finite-variable description so that it gradually recovers the anomalous nonexponential long-time relaxation of the Griffiths phase in the presence of bond-percolation clusters, $p>p_c$; (iv) clarify the relation between the finite-variable approximation and dynamical replica theory for quenched-disordered Bethe systems — whether the frameworks are complementary and how their predictions, including the special temperature where relaxation changes qualitatively, map onto each other; (v) determine how the graph-induced inhomogeneity assumption and the finite-environment approximation behave under replica-symmetry breaking, i.e. what modifications the closure requires when multiple pure states occur on tree-like graphs. Items (i) and (iii) control the error and reach of the closure, (ii) explains its central empirical success, and (iv)-(v) locate it within the existing theory of disordered dynamics; together they decide whether this reduced description is a controlled approximation or an uncontrolled coincidence.

References

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