Tomasz Gubiec

Structural and topological phase transitions on the German Stock Exchange

We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two transitions of the topology of a complex network representing the FSE were found. The first transition is from a hierarchical scale-free MST representing the stock market before the recent worldwide financial crash, to a superstar-like MST decorated by a scale-free hierarchy of trees representing the market’s state for the period containing the crash. Subsequently, a transition is observed from this transient, (meta)stable state of the crash to a hierarchical scale-free MST decorated by several star-like trees after the worldwide financial crash. The phase transitions observed are analogous to the ones we obtained earlier for the Warsaw Stock Exchange and more pronounced than those found by Onnela–Chakraborti–Kaski–Kertesz for the S&P 500 index in the vicinity of Black Monday (October 19, 1987) and also in the vicinity of January 1, 1998. Our results provide an empirical foundation for the future theory of dynamical, structural and topological phase transitions on financial markets.

Fingered growth in channel geometry: A Loewner-equation approach

A simple model of Laplacian growth is considered, in which the growth takes place only at the tips of long, thin fingers. Following Carleson and Makarov [L. Carleson and N. Makarov, J. Anal. Math. 87, 103 (2002)], the evolution of the fingers is studied with use of the deterministic Loewner equation. The method is then extended to study the growth in a linear channel with reflecting sidewalls. One- and two-finger solutions are found and analyzed. It turns out that the presence of the walls has a significant influence on the shapes of the fingers and the dynamics of the screening process, in which longer fingers suppress the growth of the shorter ones.

Stokesian dynamics of close particles

Stokesian dynamics simulations of close particles are reported, taking into account lubrication forces and many-body hydrodynamic interactions between spheres. A periodic trajectory of three particles maintaining a permanent proximity to each other has been found and analyzed. This solution is used as a benchmark to study the accuracy and stability of various numerical integration schemes. In particular, different methods of preventing unphysical overlaps of the particles are considered and potential artifacts discussed.

Intra-day variability of the stock market activity versus stationarity of the financial time series

In this paper we propose a new approach to a well-known phenomena of intra-day activity pattern on the stock market. We suggest that seasonality of inter-transaction times has a more significant impact than intra-day pattern of volatility. Our aim is not to remove the intra-day pattern from the data but to describe its impact on autocorrelation function estimators. We obtain an exact, analytical formula relating estimators of the autocorrelation functions of non-stationary (seasonal) process to its stationary counterpart. Hence, we prove that the day seasonality of inter-transaction times extends the memory of the process. That is, autocorrelation of both, price returns and their absolute values, relaxation to zero is longer.