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https://hdl.handle.net/2142/120396
Description
Title
Assorted problems in statistical mechanics
Author(s)
Krishnan, Kesav S.
Issue Date
2023-04-28
Director of Research (if dissertation) or Advisor (if thesis)
We address two distinct problems that fall under the broad umbrella of statistical mechanics, one demonstrating a phase transition and one that does not.
The first concerns the typical behavior of the discrete focusing Non Linear Schr\"odinger equation. The NLS is a famous example of a dispersive PDE that admits spatially localized, time periodic solutions called solitons. It is interesting to then study the classes of initial data lead to soliton solutions, and also those that lead to dispersive behavior. Invariant probability measures are a natural means of doing so and characterizing typicality of behavior. After surveying some known results on the invariant measure problem for the focusing NLS, we describe obstacles encountered in the cases where the dimension $d$ of the space under consideration is large, and the power of the non linearity $p$ is large as well. We discuss how spatial discretization allows us to circumvent some of these obstacles, and then analyse the thermodynamics for the focusing discrete NLS. For a naturally defined invariant Gibbs measure, we demonstrate convergence of free energy and establish the existence of a phase transition with respect to a parameter governing the strength of the non linearity. In what we refer to as the `soliton' phase, we see that typical functions are sharply peaked, with one sharply peaked region, surrounded by a background of fluctuations. In what we refer to as the dispersive `phase', there is no sharply peaked region and a typical function resembles a Gaussian Free Field conditioned to have a given $\ell^{2}-$norm. This is the main subject matter in \cite{DKK}
The second problem described in this thesis is that of the monomer dimer model. The classical monomer-dimer model refers to a well known Gibbs probability distribution on the space of matchings on a graph $G$. After surveying some known results, we present results on one of the first explorations of this model with vertex and edge weights are given by random variables. For a family of pseudo one dimensional graphs which we refer to as cylinder graphs, we prove an assortment of limit theorems for important thermodynamic quantities such as the mean free energy and they typical number of unpaired vertices. Finally, we establish a spatial limit theorem for the number of unpaired vertices in `sections' of the cylinder. This is the main subject matter in \cite{DK}.
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