The Dimensional Random Energy Model (DREM) of Campellone, Franz and Parisi (Phys. Rev. E 58, 5455 (1998)) places Derrida's Random Energy Model on a dd-dimensional lattice: each site carries MM Ising spins and each nearest-neighbor link carries independent Gaussian random energies depending on the spin configurations at its endpoints. In the mean-field limit M→∞M\to\infty the non-translation-invariant (NTI) version reproduces the REM entropy-collapse transition with no latent heat, while the symmetric translation-invariant (STI) version, which has a crystalline ground state, undergoes a first-order transition with finite latent heat. In d=1d=1, where the model was solved exactly by transfer matrices at finite MM, no transition occurs. The unresolved questions are: whether a genuine finite-temperature transition exists in dimensions d≥2d\ge2 and what the lower critical dimension is (known only to exceed one); whether the conjectured link between an underlying crystalline ground state and a genuine first-order transition with latent heat holds in finite dimension; and how the crossover correlation length scales with MM and whether it becomes truly divergent as M→∞M\to\infty for d≥2d\ge2. None of these questions has been answered: the DREM was never analyzed at finite MM beyond d=1d=1, while the broader question it instantiates remains open in glass physics.

Background

The Random Energy Model (REM) introduced by Derrida consists of 2N2^N configurations whose energies are independent Gaussian random variables of variance proportional to NN; it is exactly solvable and undergoes a freezing transition at a critical temperature TcT_c at which the entropy of the excited states collapses and the system condenses into its ground state. The transition is peculiar: the relevant order parameter jumps discontinuously as in a first-order transition, yet no latent heat is released and no susceptibility diverges. This 'random first-order' character is shared by a class of mean-field models with one-step replica symmetry breaking (1RSB) — in the glassy phase the overlap between two independent equilibrium configurations takes a finite set of values rather than a continuum — such as the pp-spin glass with p>2p>2 (spins coupled through random pp-body interactions); these models are the mean-field paradigm of the random first-order transition (RFOT) theory of the glass transition.

A natural question, addressed by Campellone, Franz and Parisi in 1998 with a purpose-built lattice model, is what happens to this transition when the model is given genuine spatial structure. They introduced the Dimensional Random Energy Model (DREM): on a dd-dimensional square lattice of side LL each site carries MM Ising spins, and every nearest-neighbor link carries a random energy E(σ,τ)E(\sigma,\tau), drawn from the Gaussian distribution P(E)=1Mπe−E2/MP(E)=\frac{1}{\sqrt{M\pi}}e^{-E^2/M} of variance M/2M/2, for each pair of spin configurations (σ,τ)(\sigma,\tau) at its endpoints; the partition function sums the Boltzmann factor of the total link energy over all spin configurations. In the non-translation-invariant (NTI) version the energy tables are redrawn independently on every link, producing a spin glass; in the translation-invariant (TI) version a single energy table is shared by all links of a given sample (for d>1d>1, following the source's stated convention, this shared table is also rotationally invariant, i.e. independent of the link direction), and in the symmetric translation-invariant (STI) version this table is symmetric, E(σ,τ)=E(τ,σ)E(\sigma,\tau)=E(\tau,\sigma), so that on a chessboard-decomposable (bipartite) lattice the system has a crystalline ground state in which every link simultaneously sits in its own minimum.

In the limit M→∞M\to\infty — a Kac-type limit of diverging interaction range in configuration space, in which mean field becomes exact — the NTI model recovers the REM freezing transition with no latent heat, while the STI model undergoes a first-order transition, at a temperature T~c\tilde T_c higher than the REM value, directly into the crystalline state, with a specific-heat discontinuity and a finite latent heat. For finite MM the authors solved the model exactly in d=1d=1 by transfer matrices — the free energy per site equals the top Lyapunov exponent (the exponential growth rate) of a product of random transfer matrices, and the correlation length follows from the gap between the two largest Lyapunov exponents — for M=4,5,6,7M=4,5,6,7. No phase transition occurs in one dimension at finite MM; instead a crossover from one-dimensional to mean-field behavior sharpens with increasing MM, with correlation-length peaks that grow with MM near the mean-field transition temperatures, suggesting but not establishing a divergence at M=∞M=\infty. The authors closed with an explicit conjecture: if the model has an underlying crystalline ground state, its transition in finite dimension becomes a genuine first-order transition with finite latent heat; without one, the transition shows no discontinuity in the first derivative of the free energy, with possible divergent correlations (as their companion short-range pp-spin studies suggested). What remains fully unresolved is the behavior at finite MM in spatial dimensions d≥2d\ge2, where no exact or controlled analysis of the DREM exists.

Problem Statement

Determine the thermodynamic behavior of the Dimensional Random Energy Model at finite interaction parameter MM in spatial dimensions d≥2d\ge2. The single scientific objective — the finite-dimensional fate of the model's mean-field transition — comprises three jointly linked questions: (i) existence and location: does a genuine finite-temperature thermodynamic phase transition occur for some finite d≥2d\ge2, and what is the lower critical dimension separating dimensions with and without a transition (a finite-MM quantity, possibly MM-dependent, since below it the transition disappears when MM is finite, while at M→∞M\to\infty the mean-field solution holds in every dimension; currently it is only known to exceed one, since the d=1d=1 model shows no transition at finite MM)? (ii) order and mechanism: what is the character of the transition in the two versions of the model — in particular, does the symmetric translation-invariant (STI) version, whose ground state is crystalline, undergo a true first-order transition with finite latent heat in finite dimension while the non-translation-invariant (NTI) version does not, as the source authors conjectured? (iii) correlation length: how does the correlation length near the crossover scale quantitatively with MM, and does it become truly divergent in the M→∞M\to\infty limit in d≥2d\ge2? All three sub-questions are jointly required for a complete resolution: each is independently refutable and settling any one of them counts as a partial resolution, but the objective is fully resolved only when the existence and location (i), the order and mechanism (ii), and the correlation-length scaling and divergence (iii) are all settled.

References

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