Generalized contour models

First order phase transitions in lattice and continuous systems: Extension of Pirogov-Sinai theory

“…While we have not yet succeeded in doing this we believe that the present work goes a certain way in that direction; see discussion at end. Other generalizations, some of them closely related to our work, have been carried out recently by various authors [11,[13][14][15]. In particular Imbrie has extended the work of Glimm, Jaffe, and Spencer on phase transitions in quantum field theories by making a powerful generalization of Pirogov-Sinai theory to such systems [7].…”

“…Also, our results on the surface tension [18] should be extendable to the present framework (see also [19]). 6) Finally, although we have not considered this case explicitly (it is studied elsewhere [7,11,13,15]), it is clear that our method allows an extension of the Pirogov-Sinai theory to continuous spin models, at least for bounded spins. For a class of ferromagnetic models of continuous bounded spins, a low temperature expansion has been given in [20].…”

“…This includes Ruelle's [4] extension of the Peierls' argument to the symmetric continuum Widom-Rowlinson model. An outstanding exception is the Pirogov-Sinai theory [5] and its extensions [6][7][8][9][10][11][12][13][14][15]. This theory applies to general lattice models of the following type: The system is described by occupation (or spin) variables which can take on a finite number of values at each site of a d-dimensional regular lattice, d^2.…”

Gibbs–non-Gibbs properties for n-vector lattice and mean-field models

“…We want to prove that for large enough times the time-evolved measure is unique. To prove this we want to use Theorem 7 from [29] which is a Pirogov-Sinai type argument for continuous models with one ground state. Let us cite their Theorem 7.…”

“…We have to use other techniques. We will use a general contour argument from [29]. Let us recall the statement.…”