23 research outputs found
Some recent (and surprising) results on interface and contact line depinning in random media
I give a brief review of results obtained recently at Ecole Normale on the
depinning transition of interfaces and contact lines using a variety of
approaches: non-local Monte Carlo algorithms, dynamical renormalization group
calculations to 2-loop order, and exact solution of an infinite-range model.Comment: 10 pages, 2 figures. Talk given at "Horizons in complex Systems"
(Messina, December 2001), to be published in Physica
Phase diagram of a Schelling segregation model
The collective behavior in a variant of Schelling's segregation model is
characterized with methods borrowed from statistical physics, in a context
where their relevance was not conspicuous. A measure of segregation based on
cluster geometry is defined and several quantities analogous to those used to
describe physical lattice models at equilibrium are introduced. This physical
approach allows to distinguish quantitatively several regimes and to
characterize the transitions between them, leading to the building of a phase
diagram. Some of the transitions evoke empirical sudden ethnic turnovers. We
also establish links with 'spin-1' models in physics. Our approach provides
generic tools to analyze the dynamics of other socio-economic systems
Monopoly Market with Externality: an Analysis with Statistical Physics and ACE
In this paper, we explore the effects of localised externalities introduced through interaction structures upon the properties of the simplest market model: the discrete choice model with a single homogeneous product and a single seller (the monopoly case). The resulting market is viewed as a complex interactive system with a communication network. Our main goal is to understand how generic properties of complex adaptive systems can enlighten our understanding of the market mechanisms when individual decisions are inter-related. To do so we make use of an ACE (Agent based Computational Economics) approach, and we discuss analogies between simulated market mechanisms and classical collective phenomena studied in Statistical Physics. More precisely, we consider discrete choice models where the agents are subject to local positive externality. We compare two extreme special cases, the McFadden (McF) and the Thurstone (TP) models. In the McF model the individuals' willingness to pay are heterogeneous, but remain fixed. In the TP model, all the agents have the same homogeneous part of willingness to pay plus an additive random (logistic) idiosyncratic characteristic. We show that these models are formally equivalent to models studied in the Physics literature, the McF case corresponding to a `Random Field Ising model' (RFIM) at zero temperature, and the TP case to an Ising model at finite temperature in a uniform (non random) external field. From the physicist's point of view, the McF and the TP models are thus quite different: they belong to the classes of, respectively,`quenched' and `annealed' disorder, which are known to lead to very different aggregate behaviour. This paper explores some consequences for market behaviour. Considering the optimisation of profit by the monopolist, we exhibit a new `first order phase transition': if the social influence is strong enough, there is a regime where, if the mean willingness to pay increases, or if the production costs decreases, the optimal solution for the monopolist jumps from a solution with a high price and a small number of buyers, to a solution with a low price and a large number of buyers.Agent-Based Computational Economics, discret choices, consumers externality, complex adaptive system, phase transition, avalanches, interactions, hysteresis.
Schelling segregation in an open city: a kinetically constrained Blume-Emery-Griffiths spin-1 system
In the 70's Schelling introduced a multi-agent model to describe the
segregation dynamics that may occur with individuals having only weak
preferences for 'similar' neighbors. Recently variants of this model have been
discussed, in particular, with emphasis on the links with statistical physics
models. Whereas these models consider a fixed number of agents moving on a
lattice, here we present a version allowing for exchanges with an external
reservoir of agents. The density of agents is controlled by a parameter which
can be viewed as measuring the attractiveness of the city-lattice. This model
is directly related to the zero-temperature dynamics of the
Blume-Emery-Griffiths (BEG) spin-1 model, with kinetic constraints. With a
varying vacancy density, the dynamics with agents making deterministic
decisions leads to a new variety of "phases" whose main features are the
characteristics of the interfaces between clusters of agents of different
types. The domains of existence of each type of interface are obtained
analytically as well as numerically. These interfaces may completely isolate
the agents leading to another type of segregation as compared to what is
observed in the original Schelling model, and we discuss its possible
socio-economic correlates.Comment: 10 pages, 7 figures, final version accepted for publication in PR
Universal interface width distributions at the depinning threshold
We compute the probability distribution of the interface width at the
depinning threshold, using recent powerful algorithms. It confirms the
universality classes found previously. In all cases, the distribution is
surprisingly well approximated by a generalized Gaussian theory of independant
modes which decay with a characteristic propagator G(q)=1/q^(d+2 zeta); zeta,
the roughness exponent, is computed independently. A functional renormalization
analysis explains this result and allows to compute the small deviations, i.e.
a universal kurtosis ratio, in agreement with numerics. We stress the
importance of the Gaussian theory to interpret numerical data and experiments.Comment: 4 pages revtex4. See also the following article cond-mat/030146
Finite average lengths in critical loop models
A relation between the average length of loops and their free energy is
obtained for a variety of O(n)-type models on two-dimensional lattices, by
extending to finite temperatures a calculation due to Kast. We show that the
(number) averaged loop length L stays finite for all non-zero fugacities n, and
in particular it does not diverge upon entering the critical regime n -> 2+.
Fully packed loop (FPL) models with n=2 seem to obey the simple relation L = 3
L_min, where L_min is the smallest loop length allowed by the underlying
lattice. We demonstrate this analytically for the FPL model on the honeycomb
lattice and for the 4-state Potts model on the square lattice, and based on
numerical estimates obtained from a transfer matrix method we conjecture that
this is also true for the two-flavour FPL model on the square lattice. We
present in addition numerical results for the average loop length on the three
critical branches (compact, dense and dilute) of the O(n) model on the
honeycomb lattice, and discuss the limit n -> 0. Contact is made with the
predictions for the distribution of loop lengths obtained by conformal
invariance methods.Comment: 20 pages of LaTeX including 3 figure
On directed interacting animals and directed percolation
We study the phase diagram of fully directed lattice animals with
nearest-neighbour interactions on the square lattice. This model comprises
several interesting ensembles (directed site and bond trees, bond animals,
strongly embeddable animals) as special cases and its collapse transition is
equivalent to a directed bond percolation threshold. Precise estimates for the
animal size exponents in the different phases and for the critical fugacities
of these special ensembles are obtained from a phenomenological renormalization
group analysis of the correlation lengths for strips of width up to n=17. The
crossover region in the vicinity of the collapse transition is analyzed in
detail and the crossover exponent is determined directly from the
singular part of the free energy. We show using scaling arguments and an exact
relation due to Dhar that is equal to the Fisher exponent
governing the size distribution of large directed percolation clusters.Comment: 23 pages, 3 figures; J. Phys. A 35 (2002) 272