1,514 research outputs found
Dynamical Probability Distribution Function of the SK Model at High Temperatures
The microscopic probability distribution function of the
Sherrington-Kirkpatrick (SK) model of spin glasses is calculated explicitly as
a function of time by a high-temperature expansion. The resulting formula to
the third order of the inverse temperature shows that an assumption made by
Coolen, Laughton and Sherrington in their recent theory of dynamics is
violated. Deviations of their theory from exact results are estimated
quantitatively. Our formula also yields explicit expressions of the time
dependence of various macroscopic physical quantities when the temperature is
suddenly changed within the high-temperature region.Comment: LaTeX, 6 pages, Figures upon request (here revised), To be published
in J. Phys. Soc. Jpn. 65 (1996) No.
Market response to external events and interventions in spherical minority games
We solve the dynamics of large spherical Minority Games (MG) in the presence
of non-negligible time dependent external contributions to the overall market
bid. The latter represent the actions of market regulators, or other major
natural or political events that impact on the market. In contrast to
non-spherical MGs, the spherical formulation allows one to derive closed
dynamical order parameter equations in explicit form and work out the market's
response to such events fully analytically. We focus on a comparison between
the response to stationary versus oscillating market interventions, and reveal
profound and partially unexpected differences in terms of transition lines and
the volatility.Comment: 14 pages LaTeX, 5 (composite) postscript figures, submitted to
Journal of Physics
Replica methods for loopy sparse random graphs
I report on the development of a novel statistical mechanical formalism for
the analysis of random graphs with many short loops, and processes on such
graphs. The graphs are defined via maximum entropy ensembles, in which both the
degrees (via hard constraints) and the adjacency matrix spectrum (via a soft
constraint) are prescribed. The sum over graphs can be done analytically, using
a replica formalism with complex replica dimensions. All known results for
tree-like graphs are recovered in a suitable limit. For loopy graphs, the
emerging theory has an appealing and intuitive structure, suggests how message
passing algorithms should be adapted, and what is the structure of theories
describing spin systems on loopy architectures. However, the formalism is still
largely untested, and may require further adjustment and refinement.Comment: 11 pages, no figures. To be published in Proceedings of The
International Meeting on High-Dimensional Data-Driven Science (HD3-2015),
Kyoto, Japan, on 14-17 December, 201
Non-equilibrium statistical mechanics of Minority Games
In this paper I give a brief introduction to a family of simple but
non-trivial models designed to increase our understanding of collective
processes in markets, the so-called Minority Games, and their non-equilibrium
statistical mathematical analysis. Since the most commonly studied members of
this family define disordered stochastic processes without detailed balance,
the canonical technique for finding exact solutions is found to be generating
functional analysis a la De Dominicis, as originally developed in the
spin-glass community.Comment: 14 pages, short review for Cergy 2002 conference proceeding
Statistical Mechanics of Recurrent Neural Networks I. Statics
A lecture notes style review of the equilibrium statistical mechanics of
recurrent neural networks with discrete and continuous neurons (e.g. Ising,
coupled-oscillators). To be published in the Handbook of Biological Physics
(North-Holland). Accompanied by a similar review (part II) dealing with the
dynamics.Comment: 49 pages, LaTe
Order-Parameter Flow in the SK Spin-Glass II: Inclusion of Microscopic Memory Effects
We develop further a recent dynamical replica theory to describe the dynamics
of the Sherrington-Kirkpatrick spin-glass in terms of closed evolution
equations for macroscopic order parameters. We show how microscopic memory
effects can be included in the formalism through the introduction of a dynamic
order parameter function: the joint spin-field distribution. The resulting
formalism describes very accurately the relaxation phenomena observed in
numerical simulations, including the typical overall slowing down of the flow
that was missed by the previous simple two-parameter theory. The advanced
dynamical replica theory is either exact or a very good approximation.Comment: same as original, but this one is TeXabl
Theory of agent-based market models with controlled levels of greed and anxiety
We use generating functional analysis to study minority-game type market
models with generalized strategy valuation updates that control the psychology
of agents' actions. The agents' choice between trend following and contrarian
trading, and their vigor in each, depends on the overall state of the market.
Even in `fake history' models, the theory now involves an effective overall bid
process (coupled to the effective agent process) which can exhibit profound
remanence effects and new phase transitions. For some models the bid process
can be solved directly, others require Maxwell-construction type
approximations.Comment: 30 pages, 10 figure
- …