Random-Interaction Ensembles

• Random-interaction ensembles are defined by incorporating correlations among matrix elements through underlying physical models, such as the Ising lattice or two-body interactions.
• They generalize classical random-matrix theory by tuning correlation strengths, bridging independent entry regimes with strongly correlated, ordered phases.
• Applications span quantum many-body systems, nuclear structure, and network science, offering insights into emergent collective phenomena and phase transitions.

Random-interaction ensembles generalize classical random-matrix theory by incorporating correlations directly derived from underlying physical interactions, often with tunable parameters. Unlike traditional ensembles with independent entries, random-interaction ensembles encode physical constraints or correlations—frequently inherited from many-body Hamiltonians or network architectures—resulting in new universality classes of spectral statistics and deep connections to phase transitions and collective phenomena in complex systems. Their construction underpins modern approaches to both quantum and classical many-body statistical physics, mesoscopic phenomena, nuclear structure, network science, and beyond.

1. Construction Principles and Prototypical Examples

The defining feature of random-interaction ensembles is the explicit incorporation of correlations among matrix elements or interactions, typically via an underlying generative physical model. A canonical example is the “interaction-correlated random matrix” class, where the entries $\{\mathcal M_{ij}\}$ of an $N \times N$ symmetric matrix are not drawn independently but generated from the Boltzmann distribution of a correlated system—in the prototypical case, a $L \times L$ two-dimensional Ising lattice at inverse temperature $\beta=1/T$ (Saberi et al., 5 Mar 2025). The procedure consists of:

• Drawing a spin configuration $\{s_{(i,j)} = \pm 1\}$ via the Boltzmann weight $P(s) \propto \exp[-\beta H(s)]$ with $H(s) = - \sum_{\langle r, r'\rangle} s_r s_{r'}$.
• Forming a matrix $\mathcal M'$ with entries $s_{(i,j)}$ and symmetrizing: $$\mathcal M = \tfrac{1}{2}(\mathcal M' + \mathcal M'^{T})$$.
• The ensemble is defined by the joint distribution over all matrix entries as dictated by the probability of each spin configuration.

Correlation structure is thus dialed by varying $\beta$ or modifying the underlying Hamiltonian (e.g., by introducing random nonlocal links and tuning their density).

Ensembles based on quantum many-body systems follow analogous logic. The Two-Body Random Ensemble (TBRE), for instance, draws two-body interaction matrix elements $V_{ij,kl}$ as independent Gaussians only within symmetry-allowed sectors, and embeds these into the many-body Hamiltonian $$H = \sum_{i<j,k<l} V_{ij,kl} a_i^\dagger a_j^\dagger a_l a_k$$, reflecting the k-body structure of physical interactions (Abramkina et al., 2011). Similarly, embedded ensembles such as EGOE($k$) generalize this to arbitrary $k$-body random Hamiltonians embedded in larger many-particle Hilbert spaces (Vyas et al., 2017, Vyas, 2010).

2. Correlation Structure and Physical Interpolation

The key innovation of random-interaction ensembles is their ability to interpolate between classical random-matrix limits (independent entries) and strongly correlated regimes. For interaction-correlated matrices based on the Ising model (Saberi et al., 5 Mar 2025):

• At infinite temperature ($\beta=0$), all spins $s_{(i,j)}$ are statistically independent: matrix entries are i.i.d., yielding the GOE/Wigner–Dyson universality.
• For $0<\beta<\beta_c$, spins exhibit finite-range correlations (correlation length $\xi(\beta)$ finite); matrix entries are short-range correlated.
• At criticality ($\beta=\beta_c$ of the Ising model), spin–spin correlations decay algebraically, $G(r) \sim r^{-1/4}$, imparting long-range power-law correlations to the matrix entries.
• Above the transition ($\beta > \beta_c$), spontaneous symmetry breaking introduces macroscopic order, observable as isolated spectral features.

The ensemble’s tunability extends to adding random nonlocal links, yielding mean-field-like behavior and restoring standard RMT universality when the density of such links is finite.

The table below summarizes core regimes in the Ising-based random-interaction ensemble:

Regime	Correlation Structure	Spectral Universality
$\beta = 0$	i.i.d. (no correlations)	Wigner–Dyson, GOE, Tracy–Widom
$0 < \beta < \beta_c$	Short-range exponential decay	Modified semicircle, GOE edge
$\beta = \beta_c$	Power law ($r^{-1/4}$) correlations	Heavy-tailed (Student–$t$ with $f=8/3$)
$\beta > \beta_c$	Long-range order (magnetized phase)	Two isolated eigenvalue “bumps” + bulk
Mean field ($q>T_c$)	Nonlocal links, mean-field order	Wigner–Dyson restored

3. Spectral Density, Universality Classes, and Scaling Laws

Random-interaction ensembles exhibit regime-dependent spectral densities and scaling properties. In the interaction-correlated matrix model:

• For $\beta \neq \beta_c$, rescaled eigenvalue density $\rho_N(\lambda)$ approaches the semicircle law, in analogy with classical GOE.
• At $\beta = \beta_c$, $\rho_N(\lambda)$ develops heavy power-law tails fit by a symmetric Student–$t$ distribution with $f=8/3$. The large-$\lambda$ decay $\rho(\lambda) \sim |\lambda|^{-11/3}$ represents a new universality class directly tied to the algebraic spin correlations at criticality (Saberi et al., 5 Mar 2025).
• In the ordered ($\beta>\beta_c$) phase, the spectrum shows a split with two “bumps” associated with the Ising magnetization, superposed on a semicircular bulk reflecting residual fluctuations.

Rescaling arguments establish the bulk independence from system size with correct normalization, while extreme value statistics diverge markedly from the Tracy–Widom paradigm in the correlated regime.

4. Extreme-Value Statistics and Physical Order Parameters

A salient achievement of these ensembles is the rigorous link between the extremes of the spectrum and physical order parameters. At criticality, the largest eigenvalue $\lambda_{\max}$ is governed by Fréchet (type-II) extreme-value statistics due to the heavy-tailed $\rho(\lambda)$:

• The Fréchet PDF with shape parameter $k = 3/8$ emerges, with $\langle \lambda_{\max} \rangle \propto L^{3/8}$, and rescaled $$\langle \tilde{\lambda}_{\max} \rangle \equiv L^{-1/2}\langle \lambda_{\max} \rangle \sim L^{-1/8}$$ (Saberi et al., 5 Mar 2025).
• This scaling is identical to the finite-size scaling of the 2D Ising magnetization, implying that $\tilde{\lambda}_{\max}$ serves as an order parameter for the ferromagnetic transition.
• In the disordered and mean-field regimes, extremes revert to Tracy–Widom and GOE limits.

This correspondence gives a route to extract both universal critical exponents and nonuniversal amplitudes from spectral data and suggests new approaches for long-standing open problems, such as constructing the exact scaling function for critical Ising magnetization.

5. Random-Interaction Ensembles in Quantum Many-Body Systems

Quantum many-body realizations include TBRE, EGOE($k$), and variational models for mesoscopic and nuclear systems (Abramkina et al., 2011, Vyas et al., 2017, Vyas, 2010). Salient features include:

• Two-body embedded ensembles generate collective observables—such as quadrupole moments, E2 transition strengths, and even rotational bands—via random two-body interactions.
• These ensembles exhibit a high probability for emergent collective phenomena (pairing, deformation, vibrational bands) not present in classical i.i.d. RMT, reflecting persistent multipole correlations.
• In nuclear structure, the dominance of specific multipole channels (notably quadrupole–quadrupole) is dynamically generated in the low-energy sector, a statistically robust consequence of the underlying random-interaction ensemble (Abramkina et al., 2011, Lei, 2015).
• Statistical analyses recover key features of mesoscopic spectra, level bunching (odd-even staggering), and spin polarization effects, with analytic variance formulas accounting for key ground-state phenomena (Vyas, 2010).

6. Broader Applications and Network Science

Generalizations of random-interaction ensembles are found in:

• Network science, where randomization with hard or soft constraints yields null models for biological or technological graphs. Ensemble sampling methods such as maximum-entropy (“Max & Sam”) and Markov Chain Monte Carlo enable construction of random-graph ensembles with prescribed topological or functional constraints (Squartini et al., 2014, Samal et al., 2010).
• Random quantum circuits and decoherence models, where embedded ensembles (fermionic EGOE, bosonic BEGOE) describe complex unitary evolution with randomized few-body interactions and have been used to analyze entanglement dynamics, qubit decoherence, and eigenstate thermalization (Vyas et al., 2017, Wick et al., 2015).
• Statistical-physics problems, including the architecture of metabolic networks constrained by biochemical and functional realism, demonstrating that many “remarkable” real-world features emerge generically under random-interaction constraints (Samal et al., 2010).

7. Significance and Outlook

Random-interaction ensembles provide a rigorous link between physical models of interacting many-body systems and new universality classes in spectral statistics. By tuning interaction parameters, these ensembles interpolate between classical RMT, strongly correlated regimes, and physically ordered phases, unifying the treatment of emergent phenomena across nuclear physics, mesoscopic systems, and complex networks.

In particular, the demonstration that spectral extremes act as physical order parameters—faithfully capturing scaling exponents and distributions—suggests novel strategies for extracting universal and nonuniversal features from complex many-body systems and for tackling unresolved problems in statistical mechanics, such as the determination of critical scaling functions in the Ising model (Saberi et al., 5 Mar 2025).

Ongoing research includes the analytical characterization of joint eigenvalue distributions in highly-correlated regimes, extensions to quantum graphs, random quantum circuits, and the exploration of constrained ensemble non-equivalence in strongly heterogeneous systems (Squartini et al., 2014). The random-interaction paradigm continues to shape our understanding of randomness, correlations, and universality in complex physical and networked systems.