R. B. Stinchcombe

Analyzing and modeling 1+1d markets

We report on a statistical analysis of the Island ECN (NASDAQ) order book. We determine the static and dynamic properties of this system, and then analyze them from a physicists viewpoint using an equivalent particle system obtained by treating orders as massive particles and price as position. We identify the fundamental dynamical processes, test existing particles models of such markets against our findings, and introduce a new model of limit order markets.

Monte Carlo Simulation and Global Optimization without Parameters

We propose a new ensemble for Monte Carlo simulations, in which each state is assigned a statistical weight

Master equation for a kinetic model of a trading market and its analytic solution

1/k

Disordered asymmetric simple exclusion process: mean-field treatment.

, where

Replica-Symmetry Breaking in the Critical Behaviour of the Random Ferromagnet

k

Stick-slip statistics for two fractal surfaces: a model for earthquakes

is the number of states with smaller or equal energy. This ensemble has robust ergodicity properties and gives significant weight to the ground state, making it effective for hard optimization problems. It can be used to find free energies at all temperatures and picks up aspects of critical behaviour (if present) without any parameter tuning. We test it on the travelling salesperson problem, the Edwards-Anderson spin glass and the triangular antiferromagnet.

Particle-hopping models of vehicular traffic: Distributions of distance headways and distance between jams

We analyze an ideal-gas-like model of a trading market with quenched random saving factors for its agents and show that the steady state income (m) distribution P(m) in the model has a power law tail with Pareto index nu exactly equal to unity, confirming the earlier numerical studies on this model. The analysis starts with the development of a master equation for the time development of P(m) . Precise solutions are then obtained in some special cases.

Non-constant rates and over-diffusive prices in a simple model of limit order markets

We provide two complementary approaches to the treatment of disorder in a fundamental nonequilibrium model, the asymmetric simple exclusion process. First, a mean-field steady-state mapping is generalized to the disordered case, where it provides a mapping of probability distributions and demonstrates how disorder results in a new flat regime in the steady-state current-density plot for periodic boundary conditions. This effect was earlier observed by Phys. Rev. E 58, 1911 (1998)] but we provide a treatment for more general distributions of disorder, including both numerical results and analytic expressions for the width 2 Delta(C) of the flat section. We then apply an argument based on moving shock fronts [Europhys. Lett. 48, 257 (1999)]] to show how this leads to an increase in the high-current region of the phase diagram for open boundary conditions. Second, we show how equivalent results can be obtained easily by taking the continuum limit of the problem and then using a disordered version of the well-known Cole-Hopf mapping to linearize the equation. Within this approach we show that adding disorder induces a localization transformation (verified by numerical scaling), and Delta(C) maps to an inverse localization length, helping to give a physical interpretation to the problem.

Disorder in non-equilibrium models

We study the critical properties of the weakly disordered p-component random Heisenberg ferromagnet. It is shown that if the specific-heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions D=4- epsilon , which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to replica-symmetry breaking (RSB) potentials such as those found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have a structure known as the one-step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.

Exact probability function for bulk density and current in the asymmetric exclusion process

Following the observations of the self-similarity in various length scales in the roughness of the fractured solid surfaces, we propose here a new model for the earthquake. We demonstrate rigorously that the contact area distribution between two fractal surfaces follows a unique power law. This is then utilised to show that the elastic energy releases for slips between two rough fractal surfaces indeed follow a Guttenberg–Richter like power law.