The application of statistical mechanics on the study of glassy behaviors in transportation networks and dynamics in models of financial markets .

摘要

In this thesis, we study two interdisciplinary problems in the framework of statistical physics, which show the broad applicability of physics on problems with various origins.;The first problem corresponds to an optimization problem in allocating resources on random regular networks. Frustrations arise from competition for resources. When the initial resources are uniform, different regimes with discrete fractions of satisfied nodes are observed, resembling the Devil's staircase. We apply the spin glass theory in analyses and demonstrate how functional recursions are converted to simple recursions of probabilities. Equilibrium properties such as the average energy and the fraction of free nodes are derived. When the initial resources are bimodally distributed, increases in the fraction of rich nodes induce a glassy transition, entering a glassy phase described by the existence of multiple metastable states, in which we employ the replica symmetry breaking ansatz for analysis.;The second problem corresponds to the study of multi-agent systems modeling financial markets. Agents in the system trade among themselves, and self-organize to produce macroscopic trading behaviors resembling the real financial markets. These behaviors include the arbitraging activities, the setting up and the following of price trends. A phase diagram of these behaviors is obtained, as a function of the sensitivity of price and the market impact factor. We finally test the applicability of the models with real financial data including the Hang Seng Index, the Nasdaq Composite and the Dow Jones Industrial Average. A substantial fraction of agents gains faster than the inflation rate of the indices, suggesting the possibility of using multi-agent systems as a tool for real trading.