Frequently, real economic agents do not follow purely rational strategies.  These individual non-rational behaviors (due to errors in judgment, incomplete information, emotional bias, etc.) can result in some fascinating organized large-scale structures, which depend on the degree of non-rational behavior.

We look at two such models for Potential Games [Shapley and Monderer]: a dynamical drift-diffusion model, and a static large deviation theory model based on Shannon information entropy and arbitrage.  The equilibrium measure in both cases is the Gibbs measure found in statistical mechanics.  We show that the variables that gauge non-rational behavior in both models are related to “temperature” by a fluctuation-dissipation relation.

A type of localized discrete Cournot oligopoly has a rich phase diagram with an "antiferromagnetic" checkerboard state, striped states and maze-like states with varying widths, and finally a "paramagnetic" unordered state. Such phases have economic implications as to how agents compete given various restrictions on how goods are distributed. 

The theory is also applied to a Speculative and Hedging Model in Oil and U.S. Dollar Markets [Carfi and Musolino] for a single multinational “airline” and many “bank” players.  Based on results for the Nash equilibrium (zero temperature) and preliminary results, there is a phase transition for which a single equilibrium exists at higher non-rational behavior (high temperature), and two equilibria at lower non-rational behavior (low temperature), when the “airline” makes no purchase of oil.  The low temperature phase is in the spirit of the Sonnenschein–Mantel–Debreu theorem, with the extra insight of symmetry-breaking to explain multiple equilibria.  Likewise, Huw Dixon’s result on the “inevitability of collusion” is shown to hold for a Cournot Oligopoly with a Veblen good.  Purely rational neoclassical theory (i.e., Nash equilibrium analysis) alone does not predict this, even though it is observed to occur in more general cases.

After defining the spectral theory of orthogonal polynomials on the unit circle (OPUC) and real line (OPRL), I'll describe Verblunsky's version of Szego's theorem as a sum rule for OPUC and the Killip--Simon sum rule for OPRL and their spectral consequences. Next I'll explain the original proof of Killip--Simon using representation theorems for meromorphic Herglotz functions. Finally I'll focus on recent work of Gambo, Nagel and Rouault who obtain the sum rules using large deviations for random matrices.

Kumura showed that there are no eigenvalues embedded in the essential
spectrum of the Laplacian on $n$-dimensional noncompact
complete Riemannian manifold $(M_n, g)$, if the radial curvature $K_{\rm
rad}+1=o(r^{-1})$ as $r$ goes to infinity.

Given any finite/countable set of positive energies $\{\lambda_n\}$, we
can
construct a Riemannian manifold with the decay order
$K_{\rm rad}+1=O(r^{-1})$/$K_{\rm rad}+1=\frac{C(r)}{r}$, where $C(r)\geq
0$ and $C(r) $ goes to infinity arbitrarily slowly, such that the
eigenvalues $\{\frac{(n-1)^2}{4}+\lambda_n\}$ are embedded in the
essential
spectrum $\sigma_{{\rm ess}}(-\Delta_g)=\left[\frac{(n-1)^2}{4},\infty
\right)$.

One of the classical questions about the evolution of a one
dimensional quantum system is the asymptotic rate of propagation of
the wave packet. It is usually captured through the notion of
transport exponents. Several methods were developed to estimate these
quantities in various models. However many authors only treated the
case of a state initially localized at a single site (in the discrete
setting). We show that some of these results can be extended to a
broad class of initial states with compact or even infinite support,
and explain what are the methods and obstacles to further
generalizations.