35 research outputs found
Higher order corrections to the effective potential close to the jamming transition in the perceptron model
We analyze the perceptron model performing a Plefka-like expansion of the
free energy. This model falls in the same universality class as hard spheres
near jamming, allowing to get exact predictions in high dimensions for more
complex systems. Our method enables to define an effective potential (or TAP
free energy), namely a coarse-grained functional depending on the contact
forces and the effective gaps between the particles. The derivation is
performed up to the third order, with a particular emphasis on the role of
third order corrections to the TAP free energy. These corrections, irrelevant
in a mean-field framework in the thermodynamic limit, might instead play a
fundamental role when considering finite-size effects. We also study the
typical behavior of the forces and we show that two kinds of corrections can
occur. The first contribution arises since the system is analyzed at a finite
distance from jamming, while the second one is due to finite-size corrections.
In our analysis, third order contributions vanish in the jamming limit, both
for the potential and the generalized forces, in agreement with the argument
proposed by Wyart and coworkers invoking isostaticity. Finally, we analyze the
scalings emerging close to the jamming line, which define a crossover regime
connecting the control parameters of the model to an effective temperature.Comment: 14 pages, 4 figure
Constraint satisfaction mechanisms for marginal stability and criticality in large ecosystems
We discuss a resource-competition model, which takes the MacArthur's model as
a platform, to unveil interesting connections with glassy features and jamming
in high dimension. This model presents two qualitatively different phases: a
"shielded" phase, where a collective and self-sustained behavior emerges, and a
"vulnerable" phase, where a small perturbation can destabilize the system and
contribute to population extinction. We first present our perspective based on
a strong similarity with continuous constraint satisfaction problems in their
convex regime. Then, we discuss the stability in terms of the computation of
the leading eigenvalue of the Hessian matrix of the free energy in the replica
space. This computation allows us to efficiently distinguish between the two
aforementioned phases and to relate high-dimensional critical ecosystems to
glassy phenomena in the low-temperature regime.Comment: Updated version with references added. 6 pages, 2 figure
Dynamical Mean-Field Theory and Aging Dynamics
Dynamical Mean-Field Theory (DMFT) replaces the many-body dynamical problem
with one for a single degree of freedom in a thermal bath whose features are
determined self-consistently. By focusing on models with soft disordered
-spin interactions, we show how to incorporate the mean-field theory of
aging within dynamical mean-field theory. We study cases with only one slow
time-scale, corresponding statically to the one-step replica symmetry breaking
(1RSB) phase, and cases with an infinite number of slow time-scales,
corresponding statically to the full replica symmetry breaking (FRSB) phase.
For the former, we show that the effective temperature of the slow degrees of
freedom is fixed by requiring critical dynamical behavior on short time-scales,
i.e. marginality. For the latter, we find that aging on an infinite number of
slow time-scales is governed by a stochastic equation where the clock for
dynamical evolution is fixed by the change of effective temperature, hence
obtaining a dynamical derivation of the stochastic equation at the basis of the
FRSB phase. Our results extend the realm of the mean-field theory of aging to
all situations where DMFT holds.Comment: 28 pages, 3 figure
The jamming transition in high dimension: an analytical study of the TAP equations and the effective thermodynamic potential
We present a parallel derivation of the Thouless-Anderson-Palmer (TAP)
equations and of an effective potential for the negative perceptron and soft
sphere models in high dimension. Both models are continuous constrained
satisfaction problems with a critical jamming transition characterized by the
same exponents. Our analysis reveals that a power expansion of the potential up
to the second order represents a successful framework to approach the jamming
line from the SAT phase (the region of the phase diagram where at least one
configuration verifies all the constraints), where the ground-state energy is
zero. An interesting outcome is that close to jamming the effective
thermodynamic potential has a logarithmic contribution, which turns out to be
dominant in a proper scaling regime. Our approach is quite general and can be
directly applied to other interesting models. Finally, we study the spectrum of
small harmonic fluctuations in the SAT phase recovering the typical scaling
below the cutoff frequency but a different behavior
characterized by a non-trivial exponent above it.Comment: 11 pages; a few typos correcte
An Introduction to the Theory of Spin Glasses
We review the main methods used to study spin glasses. In the first part, we
focus on methods for fully connected models and systems defined on a tree, such
as the replica method, the Thouless-Anderson-Palmer formalism, the cavity
method, and the dynamical mean-field theory. In the second part, we deal with
the description of low-dimensional systems, mostly in three spatial dimensions,
which are mostly studied through numerical simulations. We conclude by
mentioning some of the main open problems in the field.Comment: To appear as a chapter of the "Encyclopedia of Condensed Matter
Physics", 2nd edition (Elsevier
Loop expansion around the Bethe approximation through the -layer construction
For every physical model defined on a generic graph or factor graph, the
Bethe -layer construction allows building a different model for which the
Bethe approximation is exact in the large limit and it coincides with the
original model for . The perturbative series is then expressed by a
diagrammatic loop expansion in terms of so-called fat-diagrams. Our motivation
is to study some important second-order phase transitions that do exist on the
Bethe lattice but are either qualitatively different or absent in the
corresponding fully connected case. In this case the standard approach based on
a perturbative expansion around the naive mean field theory (essentially a
fully connected model) fails. On physical grounds, we expect that when the
construction is applied to a lattice in finite dimension there is a small
region of the external parameters close to the Bethe critical point where
strong deviations from mean-field behavior will be observed. In this region,
the expansion for the corrections diverges and it can be the starting
point for determining the correct non-mean-field critical exponents using
renormalization group arguments. In the end, we will show that the critical
series for the generic observable can be expressed as a sum of Feynman diagrams
with the same numerical prefactors of field theories. However, the contribution
of a given diagram is not evaluated associating Gaussian propagators to its
lines as in field theories: one has to consider the graph as a portion of the
original lattice, replacing the internal lines with appropriate one-dimensional
chains, and attaching to the internal points the appropriate number of
infinite-size Bethe trees to restore the correct local connectivity of the
original model
Copy number variations in healthy subjects. Case study: iPSC line CSSi005-A (3544) production from an individual with variation in 15q13.3 chromosome duplicating gene CHRNA7
CHRNA7, encoding the neuronal alpha7 nicotinic acetylcholine receptor (a7nAChR), is highly expressed in the brain, particularly in the hippocampus. It is situated in the 15q13.3 chromosome region, frequently associated with a Copy Number Variation (CNV), which causes its duplication or deletion. The clinical significance of CHRNA7 duplications is unknown so far, but there are several research data suggesting that they may be pathogenic, with reduced penetrance. We have produced an iPS cell line from a single healthy donor's fibroblasts carrying a 15q13.3 CNV, including CHRNA7 in order to study the exact role of this CNV during the neurodevelopment
Transition vitreuse et de jamming en théories de champ moyen et au-delà
La description détaillée des systèmes désordonnés et vitreux représente un défi central en physique statistique et de la matière condensée, puisqu'à ce jour il n'existe pas de théorie unique et établie permettant de comprendre ces systèmes, pourtant omniprésents.Ce travail de recherche est lié en particulier à l'étude des matériaux vitreux à basse température. Plus précisément, si l'on considère des systèmes formés par un ensemble de particules athermiques avec des interactions répulsives de portée finie, en augmentant la densité, on peut observer une transition dite d'encombrement ("jamming"). Celle-ci consiste en un blocage des degrés de liberté accompagné par une augmentation spectaculaire de la rigidité du matériau.Nous étudierons ce problème à l’aide d’une analogie formelle entre des modèles de sphères et le perceptron, un modèle théorique qui développe une transition d'encombrement et des phénomènes de frustration typiques des systèmes désordonnés.En tant que modèle en champ moyen, il permet d'obtenir des résultats analytiques précis et généralisables à des systèmes à haute dimension.L'enjeu majeur de cette étude est de reconstruire le spectre des modes de vibration et toutes les propriétés pertinentes d'une phase spécifique (correspondant au régime dit des sphères dures).Dans ce cadre, nous dériverons le potentiel effectif en fonction des paramètres d'ordre du modèle et nous montrerons qu'il est dominé à proximité du point de jamming par une interaction logarithmique non triviale, qui clarifiera le lien entre les forces de contact et les distances moyennes entre les particules, dans la région critique et au-delà.Comprendre pleinement la transition d'encombrement et les propriétés du perceptron nous permettra de faire des progrès dans plusieurs domaines reliés. En premier lieu, cela peut conduire à une théorie complète des systèmes amorphes, à la fois en dimension infinie et en dimension finie.De plus, le modèle du perceptron semble avoir un lien étroit avec des problèmes dits de Von Neumann. En effet, les systèmes biologiques et écologiques développent souvent des propriétés liées à une condition pseudo-critique en mettant en oeuvre des mécanismes d'optimisation de ressource-consommation.Est-il possible d'identifier un régime caractérisé par une brisure de symétrie? Quel serait le spectre de fluctuations d'énergie dans ces systèmes?Ce ne sont que quelques-unes des questions auxquelles nous essayerons de répondre dans cette thèse.Cependant, l'approximation de champ moyen peut parfois fournir des informationsincorrectes ou trompeuses, en particulier dans l'étude de certaines transitions de phase et la détermination des dimensions critiques inférieure et supérieure.Afin d'avoir une vue d'ensemble et pouvoir manipuler correctement des systèmes en dimension finie, dans la suite de la thèse nous discuterons comment obtenir un développement perturbatif systématique, applicable à tout modèle, à condition que ce dernier soit défini sur un réseau ou un graphe biparti.Notre motivation est en particulier liée à la possibilité d'étudier certaines transitions de phase du second ordre qui existent sur le réseau de Bethe - c'est-à-dire un réseau en arbre sans boucles dont chaque noeud a une connectivité fixe - mais qui sont qualitativement différentes ou absentes dans le modèle entièrement connecté correspondant.The detailed description of disordered and glassy systems represents an open problem in statistical physics and condensed matter. As yet, there is no single, well-established theory allowing to understand such systems. The research presented in this thesis is related in particular to the study of glassy materials in the low-temperature regime. More precisely, considering systems formed by athermal particles subject to repulsive short-range interactions, upon progressively increasing the density, a so-called jamming transition can be detected. It entails a freezing of the degrees of freedom and hence a huge increase of the material rigidity.We shall study this problem in view of a formal analogy between sphere models and the perceptron, a theoretical model undergoing a jamming transition and frustration phenomena typical of disordered systems. Being a mean-field model, it allows to obtain exact analytical results, which are generalizable to more complex high-dimensional settings.The main aim is to reconstruct the vibrational spectrum and all the relevant properties of a specific phase of the perceptron, corresponding to the hard-sphere regime.In this framework, we will derive the effective potential as a function of the gaps between and forces among the particles, and we will show that it is dominated by a non-trivial logarithmic interaction near the jamming point. This interaction in turn will clarify the relations existing between the relevant variables of the system, in the critical jamming region and beyond.Understanding the jamming transition and the perceptron properties will allow us to make progress in several related fields. First, this study could lay part of the groundwork towards a complete theory of amorphous systems, in both infinite and finite dimensions. Furthermore, the perceptron model seems to a have a close connection with the so-called Von Neumann problems. Indeed, biological and ecological systems often develop pseudo-critical properties and give rise to general mechanisms of resource-consumption optimisation.Is the identification of a broken symmetry regime possible? What would it yield in terms of the spectrum of the energy fluctuations?These are just a few questions we shall attempt to answer in this context.However, the mean-field approximation can sometimes provide wrong or misleading information, especially in studying certain phase transitions and determining the exact lower and upper critical dimensions. To have a broad perspective and correctly deal with finite-dimensional systems, in the second part of the thesis we will discuss obtaining a systematic perturbative expansion which can be applied to any model, as long as defined on a lattice or a bipartite graph.Our motivation is in particular due to the possibility of studying relevant second-order phase transitions which exist on the Bethe lattice — a lattice with a locally tree-like structure and fixed connectivity for each node — but which are qualitatively different or absent in the corresponding fully-connected version