245 research outputs found
Variational Bounds for the Generalized Random Energy Model
We compute the pressure of the random energy model (REM) and generalized
random energy model(GREM) by establishing variational upper and lower bounds.
For the upper bound, we generalize Guerra's ``broken replica symmetry
bounds",and identify the random probability cascade as the appropriate random
overlap structure for the model. For the REM the lower bound is obtained, in
the high temperature regime using Talagrand's concentration of measure
inequality, and in the low temperature regime using convexity and the high
temperature formula. The lower bound for the GREM follows from the lower bound
for the REM by induction. While the argument for the lower bound is fairly
standard, our proof of the upper bound is new.Comment: 24 page
Droplet States in the XXZ Heisenberg Chain
We consider the ground states of the ferromagnetic XXZ chain with spin up
boundary conditions in sectors with a fixed number of down spins. This forces
the existence of a droplet of down spins in the system. We find the exact
energy and the states that describe these droplets in the limit of an infinite
number of down spins. We prove that there is a gap in the spectrum above the
droplet states. As the XXZ Hamiltonian has a gap above the fully magnetized
ground states as well, this means that the droplet states (for sufficiently
large droplets) form an isolated band. The width of this band tends to zero in
the limit of infinitely large droplets. We also prove the analogous results for
finite chains with periodic boundary conditions and for the infinite chain.Comment: 50 pages, 2 figures (embedded eps files). A few descriptive
paragraphs are added plus some minor correction
A Universality Property of Gaussian Analytic Functions
We consider random analytic functions defined on the unit disk of the complex
plane as power series such that the coefficients are i.i.d., complex valued
random variables, with mean zero and unit variance. For the case of complex
Gaussian coefficients, Peres and Vir\'ag showed that the zero set forms a
determinantal point process with the Bergman kernel. We show that for general
choices of random coefficients, the zero set is asymptotically given by the
same distribution near the boundary of the disk, which expresses a universality
property. The proof is elementary and general.Comment: 7 pages. In the new version we shortened the proof. The original
arXiv submission is longer and more self-containe
Factorization properties in d-dimensional spin glasses. Rigorous results and some perspectives
In this paper we show that d-dimensional Gaussian spin glass models are
strongly stochastically stable, fulfill the Ghirlanda-Guerra identities in
distribution and the ultrametricity property.Comment: To appear in Journal of Statistical Physic
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