Interview with Steven Flores

Steven Flores
Picture. Steven Flores

Who are you?

My name is Steven Flores, and I am a postdoctoral researcher at the University of Helsinki.

What prompted you to pursue this field of research?

My research focuses on two-dimensional statistical mechanics models whose observables exhibit a conformal invariant scaling limit at criticality. Such models include percolation, Ising and Potts models, loop-gas models, various random walks, and more, and they frequently arise in material science and polymer science. Because they are complicated enough to accurately simulate real-world phenomena yet simple enough for tractable calculations to be possible, these models are both practical and fascinating to study. As such, they have attracted intense interest from physicists and mathematicians for decades. Together, this community has used rigorous probability theory, non-rigorous but convincing methods from theoretical physics, and computer simulation to contribute new results to this evolving field. More recently, two Fields Medals and a Boltzmann Medal were awarded to distinguished researchers for their contributions to this area.

I first became interested in conformal invariance of two-dimensional critical phenomena as a graduate student at the University of Michigan. Because of its rich mathematical structure, its real-world applicability, and the diverse arsenal of methods that one may use to study it, I decided to make this subject the focus of my graduate work. My graduate research concerns how clusters in percolation, Ising, and Potts models span and interconnect distant parts of the system domain to form crossing patterns and pinch-point configurations. To investigate these properties, I use Schramm–Loewner evolution (SLE), conformal field theory (CFT), and computer simulations to determine and verify formulas for probabilities of these events. Today, I continue this work as a postdoctoral researcher at the University of Helsinki.

What is this latest paper all about?

A little over 20 years ago, J Cardy used CFT to predict a formula for the probability that a percolation cluster, in the continuum limit, joins the opposite sides of a rectangle at the percolation critical point. His result is famously called 'Cardy's formula', and research to extend it to other models and polygons is ongoing. An example of such an extension is J Simmons's recent calculation of percolation crossing-probability formulas for a hexagon with free boundary conditions. J Simmons's new result is interesting because free boundary conditions, while natural to consider, seem to not arise so naturally in the analysis of crossing problems. Because CFT is non-rigorous, his results are predictions that require numerical confirmation. In this article, we confirm his predictions via computer simulation for equiangular hexagons whose side-lengths alternate from long to short.

What do you plan to do next?

Presently, we are preparing an article that uses CFT and results from our previous articles to predict explicit formulas for crossing probabilities of percolation, Ising and Potts models, and loop-gas models inside a polygon with an arbitrary even number of sides and with a fixed/free side-alternating boundary condition. A critical building block of these formulas is an SLE partition function, and in another article in preparation, we give explicit formulas for these functions in the case of rectangles, hexagons, and octagons. We also give an explicit means of calculating such functions for polygons with an arbitrary number of even sides. Much of this recent work developed out of my PhD research at the University of Michigan.

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