716 research outputs found
A chain rule for the expected suprema of Gaussian processes
The expected supremum of a Gaussian process indexed by the image of an index
set under a function class is bounded in terms of separate properties of the
index set and the function class. The bound is relevant to the estimation of
nonlinear transformations or the analysis of learning algorithms whenever
hypotheses are chosen from composite classes, as is the case for multi-layer
models
Typical entanglement of stabilizer states
How entangled is a randomly chosen bipartite stabilizer state? We show that
if the number of qubits each party holds is large the state will be close to
maximally entangled with probability exponentially close to one. We provide a
similar tight characterization of the entanglement present in the maximally
mixed state of a randomly chosen stabilizer code. Finally, we show that
typically very few GHZ states can be extracted from a random multipartite
stabilizer state via local unitary operations. Our main tool is a new
concentration inequality which bounds deviations from the mean of random
variables which are naturally defined on the Clifford group.Comment: Final version, to appear in PRA. 11 pages, 1 figur
Ultrametricity in the Edwards-Anderson Model
We test the property of ultrametricity for the spin glass three-dimensional
Edwards-Anderson model in zero magnetic field with numerical simulations up to
spins. We find an excellent agreement with the prediction of the mean
field theory. Since ultrametricity is not compatible with a trivial structure
of the overlap distribution our result contradicts the droplet theory.Comment: typos correcte
Central limit theorem for fluctuations in the high temperature region of the Sherrington-Kirkpatrick spin glass model
In a region above the Almeida-Thouless line, where we are able to control the
thermodynamic limit of the Sherrington-Kirkpatrick model and to prove replica
symmetry, we show that the fluctuations of the overlaps and of the free energy
are Gaussian, on the scale N^{-1/2}, for N large. The method we employ is based
on the idea, we recently developed, of introducing quadratic coupling between
two replicas. The proof makes use of the cavity equations and of concentration
of measure inequalities for the free energy.Comment: 18 page
Numerical estimate of finite size corrections to the free energy of the SK model using Guerra--Toninelli interpolation
I use an interpolating formula introduced by Guerra and Toninelli to
investigate numerically the finite size corrections to the free energy of the
Sherrington--Kirkpatrick model. The results are compatible with a behavior at , as predicted by Parisi, Ritort and Slanina, and
a behavior below
Replica bounds for diluted non-Poissonian spin systems
In this paper we extend replica bounds and free energy subadditivity
arguments to diluted spin-glass models on graphs with arbitrary, non-Poissonian
degree distribution. The new difficulties specific of this case are overcome
introducing an interpolation procedure that stresses the relation between
interpolation methods and the cavity method. As a byproduct we obtain
self-averaging identities that generalize the Ghirlanda-Guerra ones to the
multi-overlap case.Comment: Latex file, 15 pages, 2 eps figures; Weak point revised and
corrected; Misprints correcte
New technique for replica symmetry breaking with application to the SK-model at and near T=0
We describe a novel method which allows the treatment of high orders of
replica-symmetry-breaking (RSB) at low temperatures as well as at T=0 directly,
without a need for approximations or scaling assumptions. It yields the low
temperature order function q(a,T) in the full range and is
complete in the sense that all observables can be calculated from it. The
behavior of some observables and the finite RSB theory itself is analyzed as
one approaches continuous RSB. The validity and applicability of the
traditional continuous formulation is then scrutinized and a new continuous RSB
formulation is proposed
Positive temperature versions of two theorems on first-passage percolation
The estimates on the fluctuations of first-passsage percolation due to
Talagrand (a tail bound) and Benjamini--Kalai--Schramm (a sublinear variance
bound) are transcribed into the positive-temperature setting of random
Schroedinger operators.Comment: 15 pp; to appear in GAFA Seminar Note
The replica symmetric behavior of the analogical neural network
In this paper we continue our investigation of the analogical neural network,
paying interest to its replica symmetric behavior in the absence of external
fields of any type. Bridging the neural network to a bipartite spin-glass, we
introduce and apply a new interpolation scheme to its free energy that
naturally extends the interpolation via cavity fields or stochastic
perturbations to these models. As a result we obtain the free energy of the
system as a sum rule, which, at least at the replica symmetric level, can be
solved exactly. As a next step we study its related self-consistent equations
for the order parameters and their rescaled fluctuations, found to diverge on
the same critical line of the standard Amit-Gutfreund-Sompolinsky theory.Comment: 17 page
The Ising-Sherrington-Kirpatrick model in a magnetic field at high temperature
We study a spin system on a large box with both Ising interaction and
Sherrington-Kirpatrick couplings, in the presence of an external field. Our
results are: (i) existence of the pressure in the limit of an infinite box.
When both Ising and Sherrington-Kirpatrick temperatures are high enough, we
prove that: (ii) the value of the pressure is given by a suitable replica
symmetric solution, and (iii) the fluctuations of the pressure are of order of
the inverse of the square of the volume with a normal distribution in the
limit. In this regime, the pressure can be expressed in terms of random field
Ising models
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