arXiv subjects

Silvio Franz

Publications and source records attributed to Silvio Franz.

At least 19 recordsLinked to original sources

arXiv2026

The Longest Increasing Subsequence Problem revisited

The Longest Increasing Subsequence problem - a classic combinatorial challenge with deep connections to statistical mechanics- exhibits a rich thermodynamic landscape. Introducing a temperature we identify two distinct energy scales: A Schottky-like crossover at T_cross and a condensation transition at T_cond, below which the number of maximum-length configurations becomes sub-exponential in system size. We also show that despite the existence of polynomial-time dynamic programming algorithms for the ground state, local Monte Carlo dynamics, after sudden quenches at low temperatures, become trapped in metastable state configurations displaying characteristic glassy signatures: two-step relaxation, persistent dynamical overlaps and aging. On the other hand, logarithmic annealing tracks equilibrium down to the ground state. These results establish that thermodynamic sparsity - not energetic barriers - can render local search dynamically intractable, positioning the LIS problem as a bridge between exactly solvable optimization and glassy spin-glass phenomenology.

cond-mat.dis-nn↗

arXiv2026

Partial annealing and pattern decorrelation in associative neural networks

Using the Hopfield model as a benchmark case, the present work focuses on the investigation of partially annealed associative neural networks, wherein neural dynamics is coupled to slowly evolving patterns within the two-temperature-two-timescale framework. This setting inherently introduces a real parameter n, reminiscent of the number of replicas in the celebrated replica trick, that tunes the separation of timescales and the effective interaction between fast (i.e. the neurons) and slow (i.e. the synapses) degrees of freedom. By adapting Guerra's interpolation to the case, we derive the free energy without relying on analytical continuation. The obtained results demonstrate that negative values of n induce a progressive decorrelation of the stored patterns, thereby effectively reducing interference, promoting orthogonal configurations and ultimately conferring to the network the maximal storage alphac=1. Numerical simulations based on a mean field Monte Carlo dynamics have been employed to confirm this scenario and prove that partial annealing restores retrieval in challenging regimes, such as in the presence of biased patterns, outperforming standard decorrelation methods. These findings underscore the notion of partial annealing as an adaptive mechanism for enhancing memory organisation and retrieval in complex systems.

cond-mat.dis-nn↗

arXiv2026

Overlap locking and non-perturbative effects in spin glasses

We study the phenomenon of the locking of the order parameter (or synchronization) in spin glasses at low temperatures. When two systems with independent disorders are coupled, their overlaps become similar. A crucial question is how this effect depends on the strength of the coupling between the two systems. Non-perturbative phenomena are present when $1 \ll \Delta H \ll N$, being $\Delta H$ the coupling Hamiltonian and $N$ the size of the system. In this intermediate-coupling region, the effect is related to finite-size free-energy corrections and to the correlations in the Dyson hierarchical spin glass, a model that mimics the physics of finite-dimensional systems. We study this phenomenon in the mean-field approach, both analytically and numerically, and we finally compute the critical exponents for finite-volume corrections in mean-field theory and for the decay of correlations in the Dyson hierarchical model.

cond-mat.dis-nn↗

arXiv2024

Small field chaos in spin glasses: universal predictions from the ultrametric tree and comparison with numerical simulations

We study the chaotic behavior of the Gibbs state of spin-glasses under the application of an external magnetic field, in the crossover region where the field intensity scales proportional to $1/\sqrt{N}$, being $N$ the system size. We show that Replica Symmetry Breaking (RSB) theory provides universal predictions for chaotic behavior: they depend only on the zero-field overlap probability function $P(q)$ and are independent of other features of the system. Using solely $P(q)$ as input we can analytically predict quantitatively the statistics of the states in a small field. In the infinite volume limit, each spin-glass sample is characterized by an infinite number of states that have a tree-like structure. We generate the corresponding probability distribution through efficient sampling using a representation based on the Bolthausen-Sznitman coalescent. In this way, we can compute quantitatively properties in the presence of a magnetic field in the crossover region, the overlap probability distribution in the presence of a small field and the degree of decorrelation as the field is increased. To test our computations, we have simulated the Bethe lattice spin glass and the 4D Edwards-Anderson model, finding in both cases excellent agreement with the universal predictions.

cond-mat.dis-nn↗

arXiv2024

Soft modes in vector spin glass models on sparse random graphs

We study numerically the Hessian of low-lying minima of vector spin glass models defined on random regular graphs. We consider the two-component (XY) and three-component (Heisenberg) spin glasses at zero temperature, subjected to the action of a randomly oriented external field. Varying the intensity of the external field, these models undergo a zero temperature phase transition from a paramagnet at high field to a spin glass at low field. We study how the spectral properties of the Hessian depend on the magnetic field. In particular, we study the shape of the spectrum at low frequency and the localization properties of low energy eigenvectors across the transition. We find that in both phases the edge of the spectral density behaves as $\lambda^{3/2}$: such a behavior rules out the presence of a diverging spin-glass susceptibility $\chi_{SG}=\langle 1/\lambda^2 \rangle$. As to low energy eigenvectors, we find that the softest eigenmodes are always localized in both phases of the two models. However, by studying in detail the geometry of low energy eigenmodes across different energy scales close to the lower edge of the spectrum, we find a different behavior for the two models at the transition: in the XY case, low energy modes are typically localized; at variance, in the Heisenberg case low-energy eigenmodes with a multi-modal structure (sort of ``delocalization'') appear at an energy scale that vanishes in the infinite size limit. These geometrically non-trivial excitations, which we call Concentrated and Delocalised Low Energy Modes (CDLEM), coexist with trivially localised excitations: we interpret their existence as a sign of critical behavior related to the onset of the spin glass phase.

cond-mat.dis-nn↗

arXiv2024

Random features and polynomial rules

Random features models play a distinguished role in the theory of deep learning, describing the behavior of neural networks close to their infinite-width limit. In this work, we present a thorough analysis of the generalization performance of random features models for generic supervised learning problems with Gaussian data. Our approach, built with tools from the statistical mechanics of disordered systems, maps the random features model to an equivalent polynomial model, and allows us to plot average generalization curves as functions of the two main control parameters of the problem: the number of random features $N$ and the size $P$ of the training set, both assumed to scale as powers in the input dimension $D$. Our results extend the case of proportional scaling between $N$, $P$ and $D$. They are in accordance with rigorous bounds known for certain particular learning tasks and are in quantitative agreement with numerical experiments performed over many order of magnitudes of $N$ and $P$. We find good agreement also far from the asymptotic limits where $D\to \infty$ and at least one between $P/D^K$, $N/D^L$ remains finite.

cond-mat.dis-nn↗

arXiv2024

Darcy's law of yield stress fluids on a treelike network

Understanding the flow of yield stress fluids in porous media is a major challenge. In particular, experiments and extensive numerical simulations report a non-linear Darcy law as a function of the pressure gradient. In this letter, we consider a tree-like porous structure for which the problem of the flow can be resolved exactly thanks to a mapping with the directed polymer (DP) with disordered bond energies on the Cayley tree. Our results confirm the non-linear behavior of the flow and expresses its full pressure-dependence via the density of low-energy paths of DP restricted to vanishing overlap. These universal predictions are confirmed by extensive numerical simulations.

cond-mat.soft↗

arXiv2022

Satisfiability transition in asymmetric neural networks

Asymmetry in the synaptic interactions between neurons plays a crucial role in determining the memory storage and retrieval properties of recurrent neural networks. In this work, we analyze the problem of storing random memories in a network of neurons connected by a synaptic matrix with a definite degree of asymmetry. We study the corresponding satisfiability and clustering transitions in the space of solutions of the constraint satisfaction problem associated with finding synaptic matrices given the memories. We find, besides the usual SAT/UNSAT transition at a critical number of memories to store in the network, an additional transition for very asymmetric matrices, where the competing constraints (definite asymmetry vs. memories storage) induce enough frustration in the problem to make it impossible to solve. This finding is particularly striking in the case of a single memory to store, where no quenched disorder is present in the system.

cond-mat.dis-nn↗

arXiv2022

The relation between Parisi scheme and multi-thermalized dynamics in finite dimensions

In this note we summarize the connections between equilibrium and slow out of equilibrium dynamics in finite dimensional glasses, such as we understand them today. If we assume that a finite-dimensional system is stable with respect to a family of weak random perturbations (stochastic stability), then its dynamics have a `Multithermalization' structure if and only if the Boltzmann-Gibbs distribution obeys an Ultrametric Parisi distribution.

cond-mat.dis-nn↗

arXiv2022

Low energy excitations of mean-field glasses

We study the linear excitations around typical energy minima of a mean-field disordered model with continuous degrees of freedom undergoing a Random First Order Transition (RFOT). Contrary to naive expectations, the spectra of linear excitations are ungapped and we find the presence of a pseudogap corresponding to localized excitations with arbitrary low excitation energy. Moving to deeper minima in the landscape, the excitations appear increasingly localized while their abundance decreases. Beside typical minima, there also exist rare ultra-stable minima, with an energy gap and no localised excitations.

cond-mat.dis-nn↗

arXiv2021

Delocalization transition in low energy excitation modes of vector spin glasses

We study the energy minima of the fully-connected $m$-components vector spin glass model at zero temperature in an external magnetic field for $m\ge 3$. The model has a zero temperature transition from a paramagnetic phase at high field to a spin glass phase at low field. We study the eigenvalues and eigenvectors of the Hessian in the minima of the Hamiltonian. The spectrum is gapless both in the paramagnetic and in the spin glass phase, with a pseudo-gap behaving as $λ^{m-1}$ in the paramagnetic phase and as $\sqrtλ$ in the spin glass phase. Despite the long-range nature of the model, the eigenstates close to the edge of the spectrum display quasi-localization properties. We show that the paramagnetic to spin glass transition corresponds to delocalization of the edge eigenvectors. We solve the model by the cavity method in the thermodynamic limit. We also perform numerical minimization of the Hamiltonian for $N\le 2048$ and compute the spectral properties, that show very strong corrections to the asymptotic scaling approaching the critical point.

cond-mat.dis-nn↗

arXiv2020

Surfing on minima of isostatic landscapes: avalanches and unjamming transition

Recently, we showed that optimization problems, both in infinite as well as in finite dimensions, for continuous variables and soft excluded volume constraints, can display entire isostatic phases where local minima of the cost function are marginally stable configurations endowed with non-linear excitations [1,2]. In this work we describe an athermal adiabatic algorithm to explore with continuity the corresponding rough high-dimensional landscape. We concentrate on a prototype problem of this kind, the spherical perceptron optimization problem with linear cost function (hinge loss). This algorithm allows to "surf" between isostatic marginally stable configurations and to investigate some properties of such landscape. In particular we focus on the statistics of avalanches occurring when local minima are destabilized. We show that when perturbing such minima, the system undergoes plastic rearrangements whose size is power law distributed and we characterize the corresponding critical exponent. Finally we investigate the critical properties of the unjamming transition, showing that the linear interaction potential gives rise to logarithmic behavior in the scaling of energy and pressure as a function of the distance from the unjamming point. For some quantities, the logarithmic corrections can be gauged out. This is the case of the number of soft constraints that are violated as a function of the distance from jamming which follows a non-trivial power law behavior.

cond-mat.dis-nn↗

arXiv2020

Gradient descent dynamics in the mixed $p$-spin spherical model: finite size simulation and comparison with mean-field integration

We perform numerical simulations of a long-range spherical spin glass with two and three body interaction terms. We study the gradient descent dynamics and the inherent structures found after a quench from initial conditions, well thermalized at temperature $T_{in}$. In large systems, the dynamics strictly agrees with the integration of the mean-field dynamical equations. In particular, we confirm the existence of an onset initial temperature, within the liquid phase, below which the energy of the inherent structures undoubtedly depends on $T_{in}$. This behavior is in contrast with that of pure models, where there is a 'threshold energy' that attracts all the initial configurations in the liquid. Our results strengthen the analogy between mean-field spin glass models and supercooled liquids.

cond-mat.dis-nn↗

arXiv2020

Critical energy landscape of linear soft spheres

We show that soft spheres interacting with a linear ramp potential when overcompressed beyond the jamming point fall in an amorphous solid phase which is critical, mechanically marginally stable and share many features with the jamming point itself. In the whole phase, the relevant local minima of the potential energy landscape display an isostatic contact network of perfectly touching spheres whose statistics is controlled by an infinite lengthscale. Excitations around such energy minima are non-linear, system spanning, and characterized by a set of non-trivial critical exponents. We perform numerical simulations to measure their values and show that, while they coincide, within numerical precision, with the critical exponents appearing at jamming, the nature of the corresponding excitations is richer. Therefore, linear soft spheres appear as a novel class of finite dimensional systems that self-organize into new, critical, marginally stable, states.

cond-mat.dis-nn↗

arXiv2019

Rethinking mean-field glassy dynamics and its relation with the energy landscape: the awkward case of the spherical mixed p-spin model

The spherical p-spin model is not only a fundamental model in statistical mechanics of disordered system, but has recently gained popularity since many hard problems in machine learning can be mapped on it. Thus the study of the out of equilibrium dynamics in this model is interesting both for the glass physics and for its implications on algorithms solving NP-hard problems. We revisit the long-time limit of the out of equilibrium dynamics of mean-field spherical mixed p-spin models. We consider quenches (gradient descent dynamics) starting from initial conditions thermalized at some temperature in the ergodic phase. We perform numerical integration of the dynamical mean-field equations of the model and we find an unexpected dynamical phase transition. Below an onset temperature, higher than the dynamical transition temperature, the asymptotic energy goes below the "threshold energy" of the dominant marginal minima of the energy function and memory of the initial condition is kept. This behavior, not present in the pure spherical p-spin model, resembles closely the one observed in simulations of glass-forming liquids. We then investigate the nature of the asymptotic dynamics, finding an aging solution that relaxes towards deep marginal minima, evolving on a restricted marginal manifold. Careful analysis, however, rules out simple aging solutions. We compute the constrained complexity in the aim of connecting the asymptotic solution to the energy landscape.

cond-mat.dis-nn↗

arXiv2019

Large deviations of glassy effective potentials

The theory of glassy fluctuations can be formulated in terms of disordered effective potentials. While the properties of the average potentials are well understood, the study of the fluctuations has been so far quite limited. Close to the MCT transition, fluctuations induced by the dynamical heterogeneities in supercooled liquids can be described by a cubic field theory in presence of a random field term. In this paper we set up the general problem of the large deviations going beyond the assumption of the vicinity to $T_{MCT}$ and analyze it in the paradigmatic case of spherical ($p$-spin) glass models. This tool can be applied to study the probability of the observation of a dynamics with memory of the initial condition in regimes where, typically, the correlation $C(t,0)$ decays to zero at long times, at finite $T$ and at $T=0$.

cond-mat.dis-nn↗

arXiv2019

Critical jammed phase of the linear perceptron

Criticality in statistical physics naturally emerges at isolated points in the phase diagram. Jamming of spheres is not an exception: varying density, it is the critical point that separates the unjammed phase where spheres do not overlap and the jammed phase where they cannot be arranged without overlaps. The same remains true in more general constraint satisfaction problems with continuous variables (CCSP) where jamming coincides with the (protocol dependent) satisfiability transition point. In this work we show that by carefully choosing the cost function to be minimized, the region of criticality extends to occupy a whole region of the jammed phase. As a working example, we consider the spherical perceptron with a linear cost function in the unsatisfiable (UNSAT) jammed phase and we perform numerical simulations which show critical power laws emerging in the configurations obtained minimizing the linear cost function. We develop a scaling theory to compute the emerging critical exponents.

cond-mat.dis-nn↗

arXiv2019

Impact of jamming criticality on low-temperature anomalies in structural glasses

We present a novel mechanism for the anomalous behaviour of the specific heat in low-temperature amorphous solids. The analytic solution of a mean-field model belonging to the same universality class as high-dimensional glasses, the spherical perceptron, suggests that there exists a crossover temperature above which the specific heat scales linearly with temperature while below it a cubic scaling is displayed. This relies on two crucial features of the phase diagram: (i) The marginal stability of the free-energy landscape, which induces a gapless phase responsible for the emergence of a power-law scaling (ii) The vicinity of the classical jamming critical point, as the crossover temperature gets lowered when approaching it. This scenario arises from a direct study of the thermodynamics of the system in the quantum regime, where we show that, contrary to crystals, the Debye approximation does not hold.

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