Currently, I'm a graduate student at the Physics Department of the University of Chicago. My interests lie within the areas of statistical physics and biophysics. I am working under the supervision of Leo Kadanoff and Philippe Cluzel.

I am interested in studying the relationship between network topology, dynamics and evolution. I explore possible evolutionary advantages of such features, like the scale-free distribution.
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I am also interested in the growth properties of Schramm Loewner Evolutions. SLE is a stochastic process which produces random growing curves in 2 dimensions. I currently study the local and global properties of such growth when SLE is driven by Levy distributed noise.
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Evolvable by design

Even the simplest biological system contains an astounding network of molecular interactions. These elementary processes are linked together in order to control and perform vital cellular processes under a wide range of environmental conditions. Such intricate systems have emerged through an evolutionary process, based on random mutative events and selective pressure. As modern biology unveils the architecture of a growing number networks, their intricate design raises a general question: Is there a specific network topology that confers some kind of evolutionary advantage?

We can demonstrate how the architecture of such systems, i.e. the way their components are organized, governs their ability to evolve. We compare two distinct designs: homogeneous networks which are randomly structured and heterogeneous scale-free networks for which the distribution of connections follows a power law.

Our simulations show that populations containing these scale-free networks can easily produce a number of functional variations which allow each population to evolve rapidly and smoothly towards some target function. By contrast, equivalent random networks evolve slowly, through a succession of rare fortuitous random mutations.

For systems randomly connected it is necessary to invoke a specific tuning of their connectivity in order to access the target faster, however such fine-tuning is not required for scale-free networks.

Thus, the network architecture governs the system's evolutionary pathway, which may account for the ubiquity of scale free networks in nature.

While this work was motivated by the study of large biological networks, the significance of scale-free networks in other fields such as sociology and economics indicates that these results could be also relevant beyond the scope of biology.

"Effects of topology on network evolution", Panos Oikonomou and Philippe Cluzel, Nature Physics, August 2006.

Stochastic geometry in two dimensions

In recent years, Stochastic Loewner evolution (SLE) was shown to describe the scaling limit of many critical phenomena, such as the Ising model, the Potts model, percolation, self-avoiding random walks and others. SLE is a stochastic process which produces random growing curves in two dimensions. The curves are determined using Loewner's equation:

    g_t(z)=2/(g_t(z)-ξ(t)), with g₀(z)=z.

Standard SLE is driven by a continuous Brownian motion (ξ(t)= κ^½B(t)) which produces a continuous fractal trace of singularities. The trace is either a simple or a self-intersecting curve (for κ<4 and 4<κ<8 respectively). When a self-intersecting curve encloses an entire area, this area is considered to be swallowed. The analytical properties of such curves have been thoroughly investigated. Using an iteration of infinitesimal conformal maps, we can simulate the SLE curves.

We consider a generalized SLE driven by a superposition of a Brownian motion and a stable Lévy process. The Lévy noise introduces jumps to the driving function which are power-law distributed (¹/_|x|^1+α where α defines the process). The jumps then cause the random curves to branch out and grow like trees.

We studied the short-distance properties of this generalized process. Using both analytic and numerical considerations we determined the probability that a point on the axis is swallowed by the trace. The traces show a qualitative change in behavior as κ=4 and α=1 each pass though critical values, respectively at four and one. The transition at κ=4 is quite analogous to the known transition of standard SLE. The latter phase transition, at α=1, is a new result. For the α<1 the trace grows as isolated trees, while for α>1 the trace forms a dense "forest" of curves.

"Global properties of Stochastic Loewner evolution driven by Levy processes", P. Oikonomou, I. Rushkin, I.A.Gruzberg and L.P. Kadanoff; (submitted Oct 2007).

"Stochastic Loewner evolution driven by Levy processes", I. Rushkin, P. Oikonomou, L.P. Kadanoff and I.A.Gruzberg; J. Stat. Mech. (2006) P01001.