Ising Model and Field Theory: Lattice Local Fields and Massive Scaling Limit

This thesis is devoted to the study of the local fields in the Ising model. The scaling limit of the critical Ising model is conjecturally described by Conformal Field Theory. The explicit predictions for the building blocks of the continuum theory (spin and energy density) have been rigorously established [HoSm13, CHI15]. We study how the field-theoretic description of these random fields extends beyond the critical regime of the model. Concretely, the thesis consists of two parts: The first part studies the behaviour of lattice local fields in the critical Ising model. A lattice local field is a function of a finite number of spins at microscopic distances from a given point. We study one-point functions of these fields (in particular, their asymptotics under scaling limit and conformal invariance). Our analysis, based on discrete complex analysis methods, results in explicit computations which are of interest in applications (e.g. [HKV17]). The second part considers the behaviour of the massive spin field. In the subcritical massive scaling limit regime first considered by Wu, McCoy, Tracy, and Barouch [WMTB76], we show that the correlations of the massive spin field in a bounded domain have a scaling limit. Furthermore, to this end we generalise the notions and methods of discrete complex analysis in the critical case to the massive regime, and give a new derivation of the formula for the two-point correlation in the full plane in terms of a Painlevé III transcendent.