165 research outputs found
Theta-point polymers in the plane and Schramm-Loewner evolution
We study the connection between polymers at the theta temperature on the
lattice and Schramm-Loewner chains with constant step length in the continuum.
The latter realize a useful algorithm for the exact sampling of tricritical
polymers, where finite-chain effects are excluded. The driving function
computed from the lattice model via a radial implementation of the zipper
method is shown to converge to Brownian motion of diffusivity kappa=6 for large
times. The distribution function of an internal portion of walk is well
approximated by that obtained from Schramm-Loewner chains. The exponent of the
correlation length nu and the leading correction-to scaling exponent Delta_1
measured in the continuum are compatible with nu=4/7 (predicted for the theta
point) and Delta_1=72/91 (predicted for percolation). Finally, we compute the
shape factor and the asphericity of the chains, finding surprising accord with
the theta-point end-to-end values.Comment: 8 pages, 6 figure
A Parafermionic Generalization of the Jaynes Cummings Model
We introduce a parafermionic version of the Jaynes Cummings Hamiltonian, by
coupling Fock parafermions (nilpotent of order ) to a 1D harmonic
oscillator, representing the interaction with a single mode of the
electromagnetic field. We argue that for and there is no
difference between Fock parafermions and quantum spins . We
also derive a semiclassical approximation of the canonical partition function
of the model by assuming to be small in the regime of large enough
total number of excitations , where the dimension of the Hilbert space of
the problem becomes constant as a function of . We observe in this case an
interesting behaviour of the average of the bosonic number operator showing a
single crossover between regimes with different integer values of this
observable. These features persist when we generalize the parafermionic
Hamiltonian by deforming the bosonic oscillator with a generic function
; the deformed bosonic oscillator corresponds to a specific choice
of the deformation function . In this particular case, we observe at most
crossovers in the behavior of the mean bosonic number operator,
suggesting a phenomenology of superradiance similar to the atoms Jaynes
Cummings model.Comment: to appear on J.Phys.
Soft bounds on diffusion produce skewed distributions and Gompertz growth
Constraints can affect dramatically the behavior of diffusion processes.
Recently, we analyzed a natural and a technological system and reported that
they perform diffusion-like discrete steps displaying a peculiar constraint,
whereby the increments of the diffusing variable are subject to
configuration-dependent bounds. This work explores theoretically some of the
revealing landmarks of such phenomenology, termed "soft bound". At long times,
the system reaches a steady state irreversibly (i.e., violating detailed
balance), characterized by a skewed "shoulder" in the density distribution, and
by a net local probability flux, which has entropic origin. The largest point
in the support of the distribution follows a saturating dynamics, expressed by
the Gompertz law, in line with empirical observations. Finally, we propose a
generic allometric scaling for the origin of soft bounds. These findings shed
light on the impact on a system of such "scaling" constraint and on its
possible generating mechanisms.Comment: 9 pages, 6 color figure
Counting the learnable functions of structured data
Cover's function counting theorem is a milestone in the theory of artificial
neural networks. It provides an answer to the fundamental question of
determining how many binary assignments (dichotomies) of points in
dimensions can be linearly realized. Regrettably, it has proved hard to extend
the same approach to more advanced problems than the classification of points.
In particular, an emerging necessity is to find methods to deal with structured
data, and specifically with non-pointlike patterns. A prominent case is that of
invariant recognition, whereby identification of a stimulus is insensitive to
irrelevant transformations on the inputs (such as rotations or changes in
perspective in an image). An object is therefore represented by an extended
perceptual manifold, consisting of inputs that are classified similarly. Here,
we develop a function counting theory for structured data of this kind, by
extending Cover's combinatorial technique, and we derive analytical expressions
for the average number of dichotomies of generically correlated sets of
patterns. As an application, we obtain a closed formula for the capacity of a
binary classifier trained to distinguish general polytopes of any dimension.
These results may help extend our theoretical understanding of generalization,
feature extraction, and invariant object recognition by neural networks
Balancing building and maintenance costs in growing transport networks
The costs associated to the length of links impose unavoidable constraints to
the growth of natural and artificial transport networks. When future network
developments can not be predicted, building and maintenance costs require
competing minimization mechanisms, and can not be optimized simultaneously.
Hereby, we study the interplay of building and maintenance costs and its impact
on the growth of transportation networks through a non-equilibrium model of
network growth. We show cost balance is a sufficient ingredient for the
emergence of tradeoffs between the network's total length and transport
effciency, of optimal strategies of construction, and of power-law temporal
correlations in the growth history of the network. Analysis of empirical ant
transport networks in the framework of this model suggests different ant
species may adopt similar optimization strategies.Comment: 4 pages main text, 2 pages references, 4 figure
Beyond the storage capacity: data driven satisfiability transition
Data structure has a dramatic impact on the properties of neural networks,
yet its significance in the established theoretical frameworks is poorly
understood. Here we compute the Vapnik-Chervonenkis entropy of a kernel machine
operating on data grouped into equally labelled subsets. At variance with the
unstructured scenario, entropy is non-monotonic in the size of the training
set, and displays an additional critical point besides the storage capacity.
Remarkably, the same behavior occurs in margin classifiers even with randomly
labelled data, as is elucidated by identifying the synaptic volume encoding the
transition. These findings reveal aspects of expressivity lying beyond the
condensed description provided by the storage capacity, and they indicate the
path towards more realistic bounds for the generalization error of neural
networks.Comment: 5 pages, 2 figure
Intermittent transport of bacterial chromosomal loci
The short-time dynamics of bacterial chromosomal loci is a mixture of
subdiffusive and active motion, in the form of rapid relocations with
near-ballistic dynamics. While previous work has shown that such rapid motions
are ubiquitous, we still have little grasp on their physical nature, and no
positive model is available that describes them. Here, we propose a minimal
theoretical model for loci movements as a fractional Brownian motion subject to
a constant but intermittent driving force, and compare simulations and
analytical calculations to data from high-resolution dynamic tracking in E.
coli. This analysis yields the characteristic time scales for intermittency.
Finally, we discuss the possible shortcomings of this model, and show that an
increase in the effective local noise felt by the chromosome associates to the
active relocations.Comment: 8 pages, 6 figures; typos added, introduction expanded, conclusions
unchange
Generalization from correlated sets of patterns in the perceptron
Generalization is a central aspect of learning theory. Here, we propose a
framework that explores an auxiliary task-dependent notion of generalization,
and attempts to quantitatively answer the following question: given two sets of
patterns with a given degree of dissimilarity, how easily will a network be
able to "unify" their interpretation? This is quantified by the volume of the
configurations of synaptic weights that classify the two sets in a similar
manner. To show the applicability of our idea in a concrete setting, we compute
this quantity for the perceptron, a simple binary classifier, using the
classical statistical physics approach in the replica-symmetric ansatz. In this
case, we show how an analytical expression measures the "distance-based
capacity", the maximum load of patterns sustainable by the network, at fixed
dissimilarity between patterns and fixed allowed number of errors. This curve
indicates that generalization is possible at any distance, but with decreasing
capacity. We propose that a distance-based definition of generalization may be
useful in numerical experiments with real-world neural networks, and to explore
computationally sub-dominant sets of synaptic solutions
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