Aging in a spin glass with a logarithmic potential

Abstract: We consider a continuous-time random walk on the discrete two dimensional box, driven by the discrete Gaussian Free Field (DGFF) acting as potential: When at a vertex, the walk waits an exponentially distributed time with mean given by the exponential of the field times an inverse temperature parameter and then jumps to one of its neighbors uniformly at random. We prove that when the temperature is below the critical value the walk exhibits ``aging'' at a range of pre-equilibrium time scales: Observed at any such time and then again after an additional time of the same order, there is a positive probability that the walk is found within finite distance from where it was before, with this probability given asymptotically by the Generalized Arcsine Law with a temperature-dependent parameter. We show that this is a consequence of an intricate trapping mechanism which localizes the walk for periods of time which increase with the age of the system, and describe the complex structure of the underlying trapping landscape, which is intimately related to the geometry of the near-extreme level-sets of the DGFF. Altogether, this work demonstrates for the first time an Arcsine-Law aging in the case of a spin-glass-type system with a logarithmically correlated potential, throughout its glassy phase, as predicted in the physic literature.

• The paper proves that a random walk in a two-dimensional discrete Gaussian free field exhibits Generalized Arcsine Law aging for every inverse temperature β > √(2π) across a broad range of pre-equilibrium scales.
• The analysis identifies regular near-extremal trap clusters as the dominant contributors to elapsed time, while deep, wide, non-isolated, and shallow traps have asymptotically negligible dynamical impact.
• The paper shows that the rescaled clock process converges to an α/β-stable subordinator, establishing aging for a genuinely logarithmically correlated potential and confirming predictions for spin-glass universality.

Model and main result

The paper studies a continuous-time random walk $X$ on the two-dimensional torus $\mathsf{T}_N$ of side length $2N-1$, driven by the discrete Gaussian Free Field (DGFF) $h$ on $\mathsf{V}_N$ with zero boundary conditions. Conditionally on $h$, the walk jumps to each nearest neighbor at rate $\tfrac14 e^{-\beta h_x}$, so that its mean holding time at vertex $x$ is $\tau_x = e^{\beta h_x}$; equivalently, it is Glauber dynamics for the Boltzmann measure $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$ on $\mathsf{T}_N$0. The parameter $\mathsf{T}_N$1 is the inverse temperature, and the model is an instance of a random walk in the random potential $\mathsf{T}_N$2.

The central result is a proof of aging throughout the glassy phase. Writing $\mathsf{T}_N$3 and $\mathsf{T}_N$4, and defining the time scale

$\mathsf{T}_N$5

the main theorem states that for any fixed $\mathsf{T}_N$6 and any $\mathsf{T}_N$7, observed at $\mathsf{T}_N$8 and again at $\mathsf{T}_N$9, the walk is found within finite ($2N-1$0) distance of its earlier position with probability converging to the Generalized Arcsine Law:

$2N-1$1

uniformly over sequences $2N-1$2 in the stated range. The convergence is annealed, but an elementary argument upgrades it to a quenched statement holding in probability with respect to the field.

Two features distinguish this from prior aging results. First, the potential has approximately logarithmic correlations: $2N-1$3 in the bulk. Second, the aging window covers a full range of pre-equilibrium scales $2N-1$4, uniformly in $2N-1$5 on compact subintervals — the arcsine parameter depends on both $2N-1$6 and the ratio $2N-1$7, but not on $2N-1$8. The authors state that this is the first mathematical confirmation of the prediction of Carpentier and Le Doussal that logarithmically correlated potentials exhibit arcsine-law aging without artificial rescaling of field strength or observation times [carpentier2001glass].

The trapping landscape

The proof rests on a detailed classification of traps, defined through the normalized depth $2N-1$9 where $h$0. Vertices with $h$1 are $h$2-trap-vertices, which cluster due to the geometry of DGFF near-extrema. With $h$3 chosen to grow faster than any power of $h$4 but slower than any polynomial in $h$5, isolated trap vertices partition into $h$6-clusters represented by their local maxima (trap-bottoms). Trap-bottoms are further classified as:

• Deep ($h$7): too few to be reached;
• Wide (normal-depth bottoms whose cluster of comparable-height vertices has diameter exceeding $h$8): reachable only in vanishing number;
• Regular (normal-depth bottoms whose above-threshold neighborhood fits inside $h$9): the only traps relevant to the dynamics;
• Shallow and non-isolated vertices: visited but contributing negligible total time.

The landscape results are quantitative. Deep, non-isolated, and wide trap-bottoms are each shown to be $\mathsf{V}_N$0 in number w.h.p., where $\mathsf{V}_N$1 is the scale capturing how many $\mathsf{V}_N$2-clusters the walk discovers by time $\mathsf{V}_N$3. The sum of depths of shallow trap-vertices is likewise $\mathsf{V}_N$4 w.h.p. Conversely, regular traps are abundant: their count is tight at order $\mathsf{V}_N$5, bounded above and below by positive constants times this quantity, uniformly in $\mathsf{V}_N$6 and $\mathsf{V}_N$7. Moreover, the empirical joint law of relative depths within regular traps converges to a product of a Pareto-type law with density proportional to $\mathsf{V}_N$8 on $\mathsf{V}_N$9 and a probability measure $h$0 on $h$1 inherited from the extremal cluster law $h$2 of the DGFF. A crucial technical input is the $h$3 summability $h$4, ensuring that the total holding time accumulated per visited cluster is finite in the limit.

Mechanism and scaling heuristics

The mechanism reconciles two competing effects. During $h$5, the jump chain makes $h$6 steps with $h$7, visiting a fraction $h$8 of any $h$9-clustered set, essentially uniformly. Meanwhile, near-extreme statistics persist down to heights $\tfrac14 e^{-\beta h_x}$0: clusters of diameter $\tfrac14 e^{-\beta h_x}$1 exist around local maxima at height $\tfrac14 e^{-\beta h_x}$2 in numbers $\tfrac14 e^{-\beta h_x}$3, with relative-height configurations still governed by $\tfrac14 e^{-\beta h_x}$4. The critical height balancing discovery against contribution to elapsed time is

$\tfrac14 e^{-\beta h_x}$5

at which the number of clusters is exactly $\tfrac14 e^{-\beta h_x}$6, so $\tfrac14 e^{-\beta h_x}$7 such clusters are found. Exponentiating $\tfrac14 e^{-\beta h_x}$8 times this height yields precisely $\tfrac14 e^{-\beta h_x}$9; multiplying by the $x$0 returns per vertex and using summability of cluster depths reproduces the scale $x$1. Clusters with higher maxima are missed entirely; those with lower maxima contribute negligible total time. This identifies the aging window as intrinsically tied to the interplay between the diffusive exploration rate and the exponential tail of extreme-value counts — a balance specific to logarithmic correlations.

A subtle point is that although the full level set at height $x$2 has size $x$3 — larger than the count of local maxima by a linear factor coming from satellites of deeper maxima — most of these vertices lie in clusters the walk never reaches. What matters geometrically is the number of $x$4-clusters, not the cardinality of the target set.

Proof architecture

The dynamical argument follows the standard recipe for BTM-type aging, adapted to the correlated setting. Successive visits to new $x$5-normal traps define stopping times $x$6 and a clock process $x$7 recording cumulative continuous time. Three structural facts reduce the problem to the clock process: the walk is w.h.p. inside a normal trap at any deterministic time in $x$8; normal traps are pairwise $x$9 apart, so being $\tau_x = e^{\beta h_x}$0-close at two times means occupying the same trap; and revisits to previously visited normal traps before step $\tau_x = e^{\beta h_x}$1 have vanishing probability. The core analytic result is weak convergence of the rescaled clock process, after multiplication by an explicit constant $\tau_x = e^{\beta h_x}$2, to a standard $\tau_x = e^{\beta h_x}$3-stable subordinator in $\tau_x = e^{\beta h_x}$4. The classical arcsine theorem for Lévy processes then converts "no jump of the subordinator over $\tau_x = e^{\beta h_x}$5" into $\tau_x = e^{\beta h_x}$6.

The convergence proof proceeds in stages: local times at distinct vertices within one trap become exchangeable at scale $\tau_x = e^{\beta h_x}$7 via Green-function asymptotics; time spent per trap visit converges to i.i.d. copies of $\tau_x = e^{\beta h_x}$8 where $\tau_x = e^{\beta h_x}$9 is Pareto-like and $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$0; passage to $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$1 uses the $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$2 summability of $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$3; and the $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$4 limit is a triangular-array argument verifying Laplace-transform convergence to $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$5 together with Aldous tightness. Hitting-time inputs for the underlying simple random walk — exponential asymptotics for hitting far-away clustered sets, uniform over sets of size up to $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$6 — extend the corresponding estimates of Ben Arous and Černý to a wider range of $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$7.

The landscape proofs rely on pointwise estimates for DGFF near-extrema derived via the concentric (dyadic) decomposition of Biskup and Louidor. Conditioning a vertex to sit at height $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$8 reduces extremal constraints to ballot problems for a backbone random walk $\mu_N^\beta(\{x\}) \propto e^{\beta h_x}$9 conditioned on its endpoint, solved with two-barrier ballot estimates featuring perturbations of order $\mathsf{T}_N$00 and an entropic-repulsion lemma. Lower bounds on the abundance of regular traps require a truncated second-moment method combined with a multiscale Gibbs–Markov decomposition into $\mathsf{T}_N$01 independent sub-boxes, controlling the binding field $\mathsf{T}_N$02 through Gaussian oscillation bounds (Dudley entropy plus Borell–TIS), chaining with the Gaussian correlation inequality, and a maximal-correlation bound for the covariance structure of $\mathsf{T}_N$03 across boxes.

Relation to prior work and universality

Static properties of the model were already established: the scaled equilibrium measure $\mathsf{T}_N$04 converges to a Poissonian construction over the critical Liouville Quantum Gravity Measure, with Poisson–Dirichlet mass statistics when $\mathsf{T}_N$05 [BL3], and at equilibrium time scales ($\mathsf{T}_N$06) the dynamics converges to supercritical Liouville Brownian Motion with instantaneous tunneling between metastable states [cortines2018dynamical]. The present work addresses the complementary pre-equilibrium regime, showing that on the way to relaxation the system is trapped in metastable low-energy states for durations comparable to its age. Compared with the REM, $\mathsf{T}_N$07-spin SK model, and BTM — where the same Generalized Arcsine Law appears — the novelty is that the result holds for a genuinely spatially correlated, logarithmic potential, placing spin glasses with log-correlated potentials in the universality class of arcsine-law aging throughout the glassy phase.

Limitations and open questions

The paper treats only the asymmetric case $\mathsf{T}_N$08 of Glauber-type dynamics, where the jump chain remains simple random walk and holding times are independent of jump direction. The physically natural symmetric choice $\mathsf{T}_N$09, where transition rates depend on energy differences, is left open; the Durrett–Resnick subordinator technique used by Gayrard et al. for the BTM is suggested as a possible route but is not developed here. Second, the scales $\mathsf{T}_N$10, $\mathsf{T}_N$11, are the longest pre-equilibrium scales; behavior at much shorter times — where the walk explores only part of the torus and the effective potential becomes the DGFF restricted to its range, an object the authors note is not understood — remains open, in contrast to the REM where aging persists over a much wider spectrum of scales. Third, the finite-volume formulation requires tuning between system size and observation time; an infinite-volume version (using the pinned DGFF or the gradient field) would remove this artifact and match the original physical model more closely. Finally, the number of regular traps should admit a genuine scaling limit after normalization by $\mathsf{T}_N$12; the paper circumvents deriving it via tightness plus convergence of empirical distributions, noting that the full limit is a non-trivial task connected to recent work on near-extremal level sets.

Conclusion

This paper establishes Generalized Arcsine Law aging for a random walk driven by the two-dimensional DGFF, valid uniformly over a range of pre-equilibrium time scales throughout the glassy phase $\mathsf{T}_N$13. The result is obtained through a complete characterization of the trapping landscape — regular traps of depth order $\mathsf{T}_N$14 dominate, while deep, wide, non-isolated, and shallow traps are dynamically irrelevant — and convergence of the clock process to a stable subordinator. It provides the first rigorous instance of arcsine aging for a spin-glass-type system with logarithmically correlated disorder, confirming the physics prediction in this setting and reducing the remaining questions to symmetric dynamics, shorter time scales, and infinite volume.