Local Energy Statistics in Directed Polymers

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. We show that this conjecture holds true as well for directed polymers in random environment. We also show that, under certain conditions, this conjecture holds for directed polymers even if energy levels that grow moderately with the volume of the system are considered

Distribution of levels in high-dimensional random landscapes

We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to $\infty$. The random fields considered include costs of assignments, weights of Hamiltonian cycles and spanning trees, energies of directed polymers, locations of particles in the branching random walk, as well as energies in the Sherrington--Kirkpatrick and Edwards--Anderson models. The distribution of levels in all models listed above is shown to be essentially the same as in a stationary Gaussian process with regularly varying nonsummable covariance function. This type of behavior is different from the Brownian bridge-type limit known for independent or stationary weakly dependent sequences of random variables.Comment: Published in at http://dx.doi.org/10.1214/11-AAP772 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Onsager's Conjecture

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the 3-D incompressible Euler equations belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve kinetic energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which do not conserve kinetic energy. The first part, relating to conservation of kinetic energy, has since been confirmed (cf. Eyink 1994, Constantin-E-Titi 1994). The second part, relating to the existence of non-conservative solutions, remains an open conjecture and is the subject of this dissertation. In groundbreaking work of De Lellis and Székelyhidi Jr. (2012), the authors constructed the first examples of non-conservative Hölder continuous weak solutions to the Euler equations. The construction was subsequently improved by Isett (2012/2013), introducing many novel ideas in order to construct 1/5− Hölder continuous weak solutions with compact support in time. Adhering more closely to the original scheme of De Lellis and Székelyhidi Jr., we present a comparatively simpler construction of 1/5− Hölder continuous non-conservative weak solutions which may in addition be made to obey a prescribed kinetic energy profile. Furthermore, we extend this scheme in order to construct weak non-conservative solutions to the Euler equations whose Hölder 1/3− norm is Lebesgue integrable in time. The dissertation will be primarily based on three papers, two of which being in collaboration with De Lellis and Székelyhidi Jr