65,940 research outputs found
Phase Transition and Strong Predictability
The statistical mechanical interpretation of algorithmic information theory
(AIT, for short) was introduced and developed in our former work [K. Tadaki,
Local Proceedings of CiE 2008, pp.425-434, 2008], where we introduced the
notion of thermodynamic quantities into AIT. These quantities are real
functions of temperature T>0. The values of all the thermodynamic quantities
diverge when T exceeds 1. This phenomenon corresponds to phase transition in
statistical mechanics. In this paper we introduce the notion of strong
predictability for an infinite binary sequence and then apply it to the
partition function Z(T), which is one of the thermodynamic quantities in AIT.
We then reveal a new computational aspect of the phase transition in AIT by
showing the critical difference of the behavior of Z(T) between T=1 and T<1 in
terms of the strong predictability for the base-two expansion of Z(T).Comment: 5 pages, LaTeX2e, no figure
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
Modeling interest rate dynamics: an infinite-dimensional approach
We present a family of models for the term structure of interest rates which
describe the interest rate curve as a stochastic process in a Hilbert space. We
start by decomposing the deformations of the term structure into the variations
of the short rate, the long rate and the fluctuations of the curve around its
average shape. This fluctuation is then described as a solution of a stochastic
evolution equation in an infinite dimensional space. In the case where
deformations are local in maturity, this equation reduces to a stochastic PDE,
of which we give the simplest example. We discuss the properties of the
solutions and show that they capture in a parsimonious manner the essential
features of yield curve dynamics: imperfect correlation between maturities,
mean reversion of interest rates and the structure of principal components of
term structure deformations. Finally, we discuss calibration issues and show
that the model parameters have a natural interpretation in terms of empirically
observed quantities.Comment: Keywords: interest rates, stochastic PDE, term structure models,
stochastic processes in Hilbert space. Other related works may be retrieved
on http://www.eleves.ens.fr:8080/home/cont/papers.htm
Quenched Randomness at First-Order Transitions
A rigorous theorem due to Aizenman and Wehr asserts that there can be no
latent heat heat in a two-dimensional system with quenched random impurities.
We examine this result, and its possible extensions to higher dimensions, in
the context of several models. For systems whose pure versions undergo a strong
first-order transition, we show that there is an asymptotically exact mapping
to the random field Ising model, at the level of the interface between the
ordered and disordered phases. This provides a physical explanation for the
above result and also implies a correspondence between the problems in higher
dimensions, including scaling relations between their exponents. The particular
example of the q-state Potts model in two dimensions has been considered in
detail by various authors and we review the numerical results obtained for this
case. Turning to weak, fluctutation-driven first-order transitions, we describe
analytic renormalisation group calculations which show how the continuous
nature of the transition is restored by randomness in two dimensions.Comment: Invited talk to be presented at STATPHYS 20, Paris, July 1998; 12
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