Upcoming Seminars & Events
Perfect matchings of Z^2 (also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves at large distances like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance. As soon as dimers mutually interact, via e.g. a local energy function favoring the alignment among neighboring dimers, the model is not solvable anymore and the dimer-dimer correlation functions decay polynomially at infinity with a non-universal (interaction-dependent) critical exponent. We prove that, nevertheless, the height fluctuations remain gaussian even in the presence of interactions, in the sense that all their moments converge to the gaussian ones at large distances. The proof is based on a combination of multiscale methods with the path-independence properties of the height function. Joint work with V. Mastropietro and F. Toninelli.
While there are four commonly observed states of matter (solid crystal, liquid, gas, and plasma), we have known for some time now that there exist many other forms of matter. For example, both quasicrystals and liquid crystals are states of matter that possess properties that are intermediate between those of crystals and conventional liquids. The focus of my talk will be disordered hyperuniform many-body systems , which can be regarded to be new distinguishable states of disordered matter in that they behave more like crystals or quasicrystals in the manner in which they suppress large-scale density fluctuations, and yet are also like liquids and glasses because they are statistically isotropic structures with no Bragg peaks. Thus, disordered hyperuniform systems can be regarded to possess a "hidden order" that is not apparent on short length scales while being structurally rotationally invariant. I will describe a variety of different examples of such disordered states of matter, both equilibrium and nonequilibrium varieties. I will demonstrate that there exist classical ground states that are hyperuniform and disordered in a high-density regime down to some critical density, below which the system undergoes a phase transition to ordered states . Disordered hyperuniform systems appear to be endowed with novel physical properties, including complete photonic band gaps comparable in size to those in photonic crystals  and improved electronic band-gap properties. Moreover, we have recently shown that photoreceptor cell patterns (responsible for detecting light) in avian retina have evolved to be disordered and hyperuniform .
1. S. Torquato and F. H. Stillinger, "Local Density Fluctuations,
Hyperuniform Systems, and Order Metrics," Phys. Rev. E, 68, 041113 (2003).
2. R. D. Batten, F. H. Stillinger and S. Torquato, "Classical Disordered
Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous
Materials," J. Appl. Phys., 104, 033504 (2008).
3. M. Florescu, S. Torquato and P. J. Steinhardt, "Designer Disordered
Materials with Large, Complete Photonic Band Gaps," Proc. Nat. Acad. Sci.,
106, 20658 (2009).
4. Y. Jiao, T. Lau, H. Haztzikirou, M. Meyer-Hermann, J. C. Corbo, and S.
Torquato, "Avian Photoreceptor Patterns Represent a Disordered
Hyperuniform Solution to a Multiscale Packing Problem," Phys. Rev. E, 89,
I will discuss some new hypothesis testing problems on random graph models. A popular example is the problem of detecting community structure in a graph. Here we will consider more exotic situations, such as testing one of the basic assumption in social network analysis: whether the graph has "geometric structure". We will also talk about dynamical models of random graphs (such as preferential attachment), and how to test different hypotheses on the "history" of these graphs.