arXiv subjects

Giorgio Parisi

Publications and source records attributed to Giorgio Parisi.

At least 109 records · Page 6Linked to original sources

arXiv2014

Finite size correction to the spectrum of regular random graphs: an analytical solution

We develop a thorough analytical study of the $O(1/N)$ correction to the spectrum of regular random graphs with $N \rightarrow \infty$ nodes. The finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the $O(1/N)$ correction to the continuous part of the spectrum. The comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement, confirming the correctness of our expression.

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arXiv2014

Calorimetric glass transition in a mean field theory approach

The study of the properties of glass-forming liquids is difficult for many reasons. Analytic solutions of mean field models are usually available only for systems embedded in a space with an unphysically high number of spatial dimensions; on the experimental and numerical side, the study of the properties of metastable glassy states requires to thermalize the system in the supercooled liquid phase, where the thermalization time may be extremely large. We consider here an hard-sphere mean field model which is solvable in any number of spatial dimensions; moreover we easily obtain thermalized configurations even in the glass phase. We study the three dimensional version of this model and we perform Monte Carlo simulations which mimic heating and cooling experiments performed on ultra-stable glasses. The numerical findings are in good agreement with the analytical results and qualitatively capture the features of ultra-stable glasses observed in experiments.

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arXiv2014

Highly optimized simulations on single- and multi-GPU systems of 3D Ising spin glass

We present a highly optimized implementation of a Monte Carlo (MC) simulator for the three-dimensional Ising spin-glass model with bimodal disorder, i.e., the 3D Edwards-Anderson model running on CUDA enabled GPUs. Multi-GPU systems exchange data by means of the Message Passing Interface (MPI). The chosen MC dynamics is the classic Metropolis one, which is purely dissipative, since the aim was the study of the critical off-equilibrium relaxation of the system. We focused on the following issues: i) the implementation of efficient access patterns for nearest neighbours in a cubic stencil and for lagged-Fibonacci-like pseudo-Random Numbers Generators (PRNGs); ii) a novel implementation of the asynchronous multispin-coding Metropolis MC step allowing to store one spin per bit and iii) a multi-GPU version based on a combination of MPI and CUDA streams. We highlight how cubic stencils and PRNGs are two subjects of very general interest because of their widespread use in many simulation codes. Our code best performances ~3 and ~5 psFlip on a GTX Titan with our implementations of the MINSTD and MT19937 respectively.

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arXiv2014

Exact theory of dense amorphous hard spheres in high dimension. III. The full RSB solution

In the first part of this paper, we derive the general replica equations that describe infinite-dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai, and Wolynes is realized in a strong sense in the mean field limit. We also suggest how the computation could be generalized in an approximate way to finite dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that the fullRSB solution predicts correctly that jammed packings are isostatic, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. We show that these predictions compare very well with numerical results.

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arXiv2014

Hopping and the Stokes-Einstein relation breakdown in simple glass formers

One of the most actively debated issues in the study of the glass transition is whether a mean-field description is a reasonable starting point for understanding experimental glass formers. Although the mean-field theory of the glass transition -- like that of other statistical systems -- is exact when the spatial dimension $d\rightarrow\infty$, the evolution of systems properties with $d$ may not be smooth. Finite-dimensional effects could dramatically change what happens in physical dimensions, $d=2,3$. For standard phase transitions finite-dimensional effects are typically captured by renormalization group methods, but for glasses the corrections are much more subtle and only partially understood. Here, we investigate hopping between localized cages formed by neighboring particles in a model that allows to cleanly isolate that effect. By bringing together results from replica theory, cavity reconstruction, void percolation, and molecular dynamics, we obtain insights into how hopping induces a breakdown of the Stokes--Einstein relation and modifies the mean-field scenario in experimental systems. Although hopping is found to supersede the dynamical glass transition, it nonetheless leaves a sizable part of the critical regime untouched. By providing a constructive framework for identifying and quantifying the role of hopping, we thus take an important step towards describing dynamic facilitation in the framework of the mean-field theory of glasses.

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arXiv2014

Replica Theory and Spin Glasses

These are notes from the lectures of Giorgio Parisi given at the autumn school "Statistical Physics, Optimization, Inference, and Message-Passing Algorithm", that took place in Les Houches, France from Monday September 30th, 2013, till Friday October 11th, 2013. The school was organized by Florent Krzakala from UPMC and ENS Paris, Federico Ricci-Tersenghi from "La Sapienza" Roma, Lenka Zdeborová from CEA Saclay and CNRS, and Riccardo Zecchina from Politecnico Torino. The first lecture contains an introduction to the replica method, along with a concrete application to the computation of the eigenvalue distribution of random matrices in the GOE. In the second lecture, the solution of the SK model is derived, along with the phenomenon of replica symmetry breaking (RSB). In the third part, the physical meaning of the RSB is explained. The ultrametricity of the space of pure states emerges as a consequence of the hierarchical RSB scheme. Moreover, it is shown how some low temperature properties of physical observables can be derived by invoking the stochastic stability principle. Lecture four contains some rigorous results on the SK model: the existence of the thermodynamic limit, and the proof of the exactness of the hierarchical RSB solution.

cond-mat.stat-mech↗

arXiv2014

Scaling hypothesis for the Euclidean bipartite matching problem

We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large $N$ limit of the average cost in dimension $d=1,2$ and of the subleading correction in higher dimension. A non-trivial scaling exponent, $γ_d=\frac{d-2}{d}$, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension $d>2$.

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arXiv2014

Finite size corrections to disordered Ising models on Random Regular Graphs

We derive the analytical expression for the first finite size correction to the average free energy of disordered Ising models on random regular graphs. The formula can be physically interpreted as a weighted sum over all non self-intersecting loops in the graph, the weight being the free-energy shift due to the addition of the loop to an infinite tree.

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arXiv2014

The crossover region between long-range and short-range interactions for the critical exponents

It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the crossover region is not well understood. In this paper we propose a general form for the crossover function and we compute it in a particular limit. We compare our predictions with the results of numerical simulations for two-dimensional long-range percolation.

cond-mat.stat-mech↗

arXiv2014

Large Deviations of Correlation Functions in Random Magnets

We present a large deviations theory of the spin-spin correlation functions in the Random Field Ising Model on the Bethe lattice, both at finite and zero temperature. Rare events of atypically correlated variables are particularly important at the critical point: the phase transition is driven by few pairs of strongly correlated spins, while the majority remains basically uncorrelated. At the zero temperature critical point the number of spin pairs correlated over a distance L is shown to be no longer exponential, but only linear in the spins separation.

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arXiv2014

Anomalous finite size corrections in random field models

The presence of a random magnetic field in ferromagnetic systems leads, in the broken phase, to an anomalous $O(\sqrt{1/N})$ convergence of some thermodynamic quantities to their asymptotic limits. Here we show a general method, based on the replica trick, to compute analytically the $O(\sqrt{1/N})$ finite size correction to the average free energy. We apply this method to two mean field Ising models, fully connected and random regular graphs, and compare the results to exact numerical algorithms. We argue that this behaviour is present in finite dimensional models as well.

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arXiv2014

Fractal free energy landscapes in structural glasses

Glasses are amorphous solids whose constituent particles are caged by their neighbors and thus cannot flow. This sluggishness is often ascribed to the free energy landscape containing multiple minima (basins) separated by high barriers. Here we show, using theory and numerical simulation, that the landscape is much rougher than is classically assumed. Deep in the glass, it undergoes a "roughness transition" to fractal basins. This brings about isostaticity at jamming and marginality of glassy states near jamming. Critical exponents for the basin width, the weak force distribution, and the spatial spread of quasi-contacts at jamming can be analytically determined. Their value is found to be compatible with numerical observations. This advance therefore incorporates the jamming transition of granular materials into the framework of glass theory. Because temperature and pressure control which features of the landscape are experienced, glass mechanics and transport are expected to reflect the features of the topology we discuss here. Hitherto mysterious properties of low-temperature glasses could be explained by this approach.

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arXiv2014

Relations between Short Range and Long Range Ising models

We perform a numerical study of the long range (LR) ferromagnetic Ising model with power law decaying interactions ($J \propto r^{-d-σ}$) both on a one-dimensional chain ($d=1$) and on a square lattice ($d=2$). We use advanced cluster algorithms to avoid the critical slowing down. We first check the validity of the relation connecting the critical behavior of the LR model with parameters $(d,σ)$ to that of a short range (SR) model in an equivalent dimension $D$. We then study the critical behavior of the $d=2$ LR model close to the lower critical $σ$, uncovering that the spatial correlation function decays with two different power laws: the effect of the subdominant power law is much stronger than finite size effects and actually makes the estimate of critical exponents very subtle. By including this subdominant power law, the numerical data are consistent with the standard renormalization group (RG) prediction by Sak, thus making not necessary (and unlikely, according to Occam's razor) the recent proposal by Picco of having a new set of RG fixed points, in addition to the mean-field one and the SR one.

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arXiv2014

Soft modes in jammed hard spheres (I): Mean field theory of the isostatic transition

In this paper we consider different models for soft modes in jammed hard spheres. We show how one can construct mean field models that can be solved analytically. The analytic solution of these models displays an excess of low energy soft modes that become more and more localized by decreasing the energy. A simple solution of these models is found in the infinite dimensional limit.

cond-mat.soft↗

arXiv2014

Ensemble renormalization group for the random field hierarchical model

The Renormalization Group (RG) methods are still far from being completely understood in quenched disordered systems. In order to gain insight into the nature of the phase transition of these systems, it is common to investigate simple models. In this work we study a real-space RG transformation on the Dyson hierarchical lattice with a random field, which led to a reconstruction of the RG flow and to an evaluation of the critical exponents of the model at T = 0. We show that this method gives very accurate estimations of the critical exponents, by comparing our results with the ones obtained by some of us using an independent method.

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arXiv2013

Spatial correlation functions and dynamical exponents in very large samples of 4D spin glasses

The study of the low temperature phase of spin glass models by means of Monte Carlo simulations is a challenging task, because of the very slow dynamics and the severe finite size effects they show. By exploiting at the best the capabilities of standard modern CPUs (especially the SSE instructions), we have been able to simulate the four-dimensional (4D) Edwards-Anderson model with Gaussian couplings up to sizes $L=70$ and for times long enough to accurately measure the asymptotic behavior. By quenching systems of different sizes to the the critical temperature and to temperatures in the whole low temperature phase, we have been able to identify the regime where finite size effects are negligible: $ξ(t) \lesssim L/7$. Our estimates for the dynamical exponent ($z \simeq 1/T$) and for the replicon exponent ($α\simeq 1.0$ and $T$-independent), that controls the decay of the spatial correlation in the zero-overlap sector, are consistent with the RSB theory, but the latter differs from the theoretically conjectured value.

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arXiv2013

Temperature chaos and quenched heterogeneities

We present a treatable generalization of the Sherrington Kirkpatrick model which introduces correlations in the elements of the coupling matrix through multiplicative disorder on the single variables and investigate the consequences on the phase diagram. We define a generalized $q_{EA}$ parameter and test the structural stability of the SK results in this correlated case evaluating the AT-line of the model. As main result we demonstrate the increase of temperature chaos effects due to heterogeneities.

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arXiv2013

Universality Classes of Critical Points in Constrained Glasses

We analyze critical points that can be induced in glassy systems by the presence of constraints. These critical points are predicted by the Mean Field Thermodynamic approach and they are precursors of the standard glass transition in absence of constraints. Through a deep analysis of the soft modes appearing in the replica field theory we can establish the universality class of these points. In the case of the "annealed potential" of a symmetric coupling between two copies of the system, the critical point is in the Ising universality class. More interestingly, is the case of the "quenched potential" where the a single copy is coupled with an equilibrium reference configuration, or the "pinned particle" case where a fraction of particles is frozen in fixed positions. In these cases we find the Random Field Ising Model (RFIM) universality class. The effective random field is a "self-generated" disorder that reflects the random choice of the reference configuration. The RFIM representation of the critical theory predicts non-trivial relations governing the leading singular behavior of relevant correlation functions, that can be tested in numerical simulations.

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