SPoDS scientific investigation addresses the statistical mechanics of systems with quenched disorder, along two connected directions: the fundamental theory of the disordered phase, and the use of disordered models to describe and exploit physical systems in which randomness is intrinsic.
1. Theory of disordered systems
Replica and cavity methods, replica symmetry breaking and renormalization group in the presence of non-perturbative randomness. Specific lines include the spin-glass phase of finite-dimensional systems in an external field, with real space renormalization and numerical simulations. Recent work extends this to the spin glass in a field at zero temperature and to long-range models, to overlap locking and non-perturbative effects, and to the lower critical dimension of the Ising spin glass, probed through low-energy excitations in elongated geometries; the lack of self-averaging in weakly disordered models is studied with the same tools. A distinct line concerns first-order and inverse phase transitions in Blume–Capel and Blume–Emery–Griffiths spin-glass models, where “freezing upon heating” occurs.
2. Glasses, dynamics and complexity
The complexity functional and the off-equilibrium dynamics of glassy systems, including two- and three-step relaxation, critical slowing down, memory, aging and the role of marginal states. Recent works establish the aging phase diagram and the exact asymptotic energies of mixed spherical spin glasses, and compute the quenched complexity of marginal states in the Sherrington–Kirkpatrick model. Related questions on stochastic processes in random environments are addressed with the same methods, from lattice random walks in Gaussian potentials to Ornstein–Uhlenbeck dynamics driven by dichotomous noise.
3. Neural networks and learning
Storage and retrieval in associative memories: the capacity of Hopfield-like models and its unexpected enhancement by spurious overlaps, consolidation through dreaming and daydreaming algorithms under bounded synaptic strength, partial annealing and pattern decorrelation, and the inference of concepts from noisy examples. Pseudo-likelihood learning yields memories that generalize even with asymmetric couplings. The same statistical-physics language extends to generative models, where the training of diffusion models exhibits critical slowing down.
4. Optimization, inference and constraint satisfaction
Hard combinatorial problems addressed with disordered-systems methods: message-passing and cluster-based optimization of Quadratic Unconstrained Binary Optimization instances, compressed sensing beyond the convex regime, transition path sampling on heterogeneous graphs, and the detection thresholds of discontinuous Baik–Ben Arous–Péché transitions. Machine-learning-enhanced Monte Carlo yields a demonstrated advantage over standard sampling in combinatorial optimization, while graph neural networks are benchmarked against hard constraint-satisfaction instances. Inverse statistical problems are studied across phase transitions, where inference becomes intrinsically harder.
5. Light in random media
Spin-glass theory of multimode and random lasers: statitical physics determination of lasing threshold, power condensation, mode-locking, and the identification of the intensity-fluctuation overlap as a glassy order parameter. The theory has been experimentally tested in organic and inorganic random lasers, including the direct measurement of the Parisi-related intensity fluctuation overlap distribution. A recent development is a smoothed-cubic spin-glass model for random lasers.
6. Photonic and quantum computation
Statistical inference on interacting nonlinear waves, phase retrieval and transmission-matrix reconstruction for light propagation through random media, with applications to imaging, focusing and reservoir computing. A current line concerns the computation of disordered models by optical means, from the analog simulation of spin-glass Monte Carlo dynamics to the multiphoton quantum simulation of generalized dense Hopfield memories on integrated photonic processors, where retrieval, spin-glass and paramagnetic phases have been experimentally observed.
7. Methods and computing
Enhanced Monte Carlo algorithms and their CPU and GPU parallelization, high-performance solvers for dynamical mean-field equations reaching long times, and message-passing and pseudo-likelihood methods for inverse problems. Cluster moves coupled to an entropic reservoir and microcanonical simulated annealing with sporadic random-number generation extend the reach of large-scale simulations, and non-convex optimization methods are developed for ground states of continuous-spin disordered models. Hard constraint-satisfaction problems are addressed with the same toolbox.