13 research outputs found
Inference-Based Decisions in a Hidden State Foraging Task:Differential Contributions of Prefrontal Cortical Areas
Fractal geometry of spin-glass models
Stability and diversity are two key properties that living entities share
with spin glasses, where they are manifested through the breaking of the phase
space into many valleys or local minima connected by saddle points. The
topology of the phase space can be conveniently condensed into a tree
structure, akin to the biological phylogenetic trees, whose tips are the local
minima and internal nodes are the lowest-energy saddles connecting those
minima. For the infinite-range Ising spin glass with p-spin interactions, we
show that the average size-frequency distribution of saddles obeys a power law
, where w=w(s) is the number of minima that can be
connected through saddle s, and D is the fractal dimension of the phase space
An investigation of the hidden structure of states in a mean field spin glass model
We study the geometrical structure of the states in the low temperature phase
of a mean field model for generalized spin glasses, the p-spin spherical model.
This structure cannot be revealed by the standard methods, mainly due to the
presence of an exponentially high number of states, each one having a vanishing
weight in the thermodynamic limit. Performing a purely entropic computation,
based on the TAP equations for this model, we define a constrained complexity
which gives the overlap distribution of the states. We find that this
distribution is continuous, non-random and highly dependent on the energy range
of the considered states. Furthermore, we show which is the geometrical shape
of the threshold landscape, giving some insight into the role played by
threshold states in the dynamical behaviour of the system.Comment: 18 pages, 8 PostScript figures, plain Te
How a spin-glass remembers. Memory and rejuvenation from intermittency data: an analysis of temperature shifts
The memory and rejuvenation aspects of intermittent heat transport are
explored theoretically and by numerical simulation for Ising spin glasses with
short-ranged interactions. The theoretical part develops a picture of
non-equilibrium glassy dynamics recently introduced by the authors. Invoking
the concept of marginal stability, this theory links irreversible
`intermittent' events, or `quakes' to thermal fluctuations of record magnitude.
The pivotal idea is that the largest energy barrier surmounted prior
to by thermal fluctuations at temperature determines the rate of the intermittent events occurring near . The idea leads
to a rate of intermittent events after a negative temperature shift given by
, where the `effective age' has
an algebraic dependence on , whose exponent contains the temperatures
before and after the shift. The analytical expression is verified by numerical
simulations. Marginal stability suggests that a positive temperature shift could erase the memory of the barrier . The simulations show
that the barrier controls the intermittent dynamics,
whose rate is hence .
Additional `rejuvenation' effects are also identified in the intermittency
data for shifts of both signs.Comment: Revised introduction and discussion. Final version to appear in
Journal of Statistical Mechanics: Theory and Experimen
Phase transition and landscape statistics of the number partitioning problem
The phase transition in the number partitioning problem (NPP), i.e., the
transition from a region in the space of control parameters in which almost all
instances have many solutions to a region in which almost all instances have no
solution, is investigated by examining the energy landscape of this classic
optimization problem. This is achieved by coding the information about the
minimum energy paths connecting pairs of minima into a tree structure, termed a
barrier tree, the leaves and internal nodes of which represent, respectively,
the minima and the lowest energy saddles connecting those minima. Here we apply
several measures of shape (balance and symmetry) as well as of branch lengths
(barrier heights) to the barrier trees that result from the landscape of the
NPP, aiming at identifying traces of the easy/hard transition. We find that it
is not possible to tell the easy regime from the hard one by visual inspection
of the trees or by measuring the barrier heights. Only the {\it difficulty}
measure, given by the maximum value of the ratio between the barrier height and
the energy surplus of local minima, succeeded in detecting traces of the phase
transition in the tree. In adddition, we show that the barrier trees associated
with the NPP are very similar to random trees, contrasting dramatically with
trees associated with the spin-glass and random energy models. We also
examine critically a recent conjecture on the equivalence between the NPP and a
truncated random energy model
Spin glass overlap barriers in three and four dimensions
For the Edwards-Anderson Ising spin-glass model in three and four dimensions
(3d and 4d) we have performed high statistics Monte Carlo calculations of those
free-energy barriers which are visible in the probability density
of the Parisi overlap parameter . The calculations rely on the
recently introduced multi-overlap algorithm. In both dimensions, within the
limits of lattice sizes investigated, these barriers are found to be
non-self-averaging and the same is true for the autocorrelation times of our
algorithm. Further, we present evidence that barriers hidden in dominate
the canonical autocorrelation times.Comment: 20 pages, Latex, 12 Postscript figures, revised version to appear in
Phys. Rev.
Metastable States in Spin Glasses and Disordered Ferromagnets
We study analytically M-spin-flip stable states in disordered short-ranged
Ising models (spin glasses and ferromagnets) in all dimensions and for all M.
Our approach is primarily dynamical and is based on the convergence of a
zero-temperature dynamical process with flips of lattice animals up to size M
and starting from a deep quench, to a metastable limit. The results (rigorous
and nonrigorous, in infinite and finite volumes) concern many aspects of
metastable states: their numbers, basins of attraction, energy densities,
overlaps, remanent magnetizations and relations to thermodynamic states. For
example, we show that their overlap distribution is a delta-function at zero.
We also define a dynamics for M=infinity, which provides a potential tool for
investigating ground state structure.Comment: 34 pages (LaTeX); to appear in Physical Review
Energy barriers in SK spin-glass model
The height of barriers separating metastable states is studied for the infinite range Ising spin-glass model. A barrier configuration corresponds to a saddle point {mi, i = 1, ..., N, — 1 ≤ mi ≤ 1} of an energy surface that smoothly interpolates the energy in the hypercube. For low energy barriers, which are relevant for the dynamics, the number of independent descending directions from the saddle point is finite when N → ∞. It is found that these descending directions are contained in the linear subspace of the hypercube [—1,1]N generated by directions for which Σ j Jij mj = 0. Numerical estimates were performed by two distinct algorithms which lead to consistent results. Assuming barrier heights growing with system size N as Nα we find α = 0.34 ± 0.08.Nous étudions les hauteurs de barrière séparant des états métastables pour le modèle de verre de spin avec symétrie d'Ising et portée infinie. Une configuration de barrière correspond à un col de {mi, i = 1, ..., N, - 1 ≤ mi ≤ 1} de la surface d'énergie qui interpole de façon régulière l'énergie dans l'hypercube. Pour des barrières d'énergie faibles, qui sont importantes pour la dynamique, le nombre de directions descendantes au col est fini dans la limite N → ∞. Nous trouvons que ces directions sont contenues dans un sous-espace linéaire de l'hypercube [—1,1]N engendré par les directions pour lesquelles Σj Jij mj = 0. Nous avons fait des simulations numeriques par deux algorithmes différents. Les résultats sont cohérents. Si nous supposons que les hauteurs de barrière croissent avec la taille N du système comme Nα, nous trouvons α = 0,34 ± 0,08