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# Large deviations for doubly indexed stochastic processes with applications to statistical mechanics

Abstract (Summary)
The theory of large deviations studies situations in which certain probabilities involving a given stochastic process decay to zero exponentially fast. One of the aims of this dissertation is to extend this theory to the setting in which the stochastic processes under consideration are indexed by two parameters, rather than the usual one parameter. The introduction of the second index often allows one to study more easily the large deviation asymptotics of processes with a spatial component. Such doubly indexed processes, interesting in their own right, are especially so because of their applications to a class of statistical mechanical models of fluid turbulence. Indeed, the powerful apparatus of large deviation theory can be applied to a general statistical mechanical model via the program outlined in Chapter 2. A second aim of this dissertation is to apply our two parameter large deviation results to a particular model of two-dimensional fluid turbulence introduced in Chapter 6. The main probabalistic theorem in the dissertation is the large deviation principle for the doubly indexed sequence of random probability measures$$W\sb{r,q}(dx\times dy)\ \doteq\ \theta (dx)\otimes\sum\sbsp{k=1}{2r}1\sb{D\sb{r,k}}(x)L\sb{q,k}(dy).$$Here $\theta$ is a probability measure on a Polish space $\chi$, $\{D\sb{r,k},\ k = 1,\...,2\sp{r}\}$ is a dyadic partition of $\chi$ (hence the use of $2\sp{r}$ summands) satisfying $\theta (D\sb{r,k}) = 1/2\sp{r},$ and $L\sb{q,1},\ L\sb{q,2},\... ,L\sb{q,2\sp{r}}$ is an independent, identically distributed sequence of random probability measures on a Polish space $\cal Y$ such that $\{L\sb{q,k},\ q\in {\rm I\!N}\}$ satisfies the large deviation principle with a convex rate function. A number of related asymptotic results are also derived. In the final two chapters of the dissertation we introduce a statistical mechanical model of two-dimensional turbulence constructed on a uniform lattice of points of the unit torus. We use a doubly indexed process closely related to $W\sb{r,q}$ to approximate the process which arises naturally in applying large deviation theory to this model. The two parameter large deviation principle for the doubly indexed process then leads to the evaluation of the asymptotics of certain key statistical mechanical quantities related to the partition function and the Gibbs states.
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