Hermite's theorem makes it known that there are three levels of mathematical frames in which a
simple addition formula is valid. They are rational q-analogue and elliptic-analogue. Based
on the addition formula and associated mathematical structures productive studies have been
carried out in the process of q-extension of the rational (classical) formulas in enumerative
combinatorics theory of special functions representation theory study of integrable systems
and so on. Originating from the paper by Date Jimbo Kuniba Miwa and Okado on the exactly
solvable statistical mechanics models using the theta function identities (1987) the formulas
obtained at the q-level are now extended to the elliptic level in many research fields in
mathematics and theoretical physics. In the present monograph the recent progress of the
elliptic extensions in the study of statistical and stochastic models in equilibrium and
nonequilibrium statistical mechanics and probability theory is shown. At the elliptic level
many special functions are used including Jacobi's theta functions Weierstrass elliptic
functions Jacobi's elliptic functions and others. This monograph is not intended to be a
handbook of mathematical formulas of these elliptic functions however. Thus use is made only
of the theta function of a complex-valued argument and a real-valued nome which is a
simplified version of the four kinds of Jacobi's theta functions. Then the seven systems of
orthogonal theta functions written using a polynomial of the argument multiplied by a single
theta function or pairs of such functions can be defined. They were introduced by Rosengren
and Schlosser (2006) in association with the seven irreducible reduced affine root systems.
Using Rosengren and Schlosser's theta functions non-colliding Brownian bridges on a
one-dimensional torus and an interval are discussed along with determinantal point processes
on a two-dimensional torus. Their scaling limitsare argued and the infinite particle systems
are derived. Such limit transitions will be regarded as the mathematical realizations of the
thermodynamic or hydrodynamic limits that are central subjects of statistical mechanics.
