Roman Kotecký

Cluster expansion for abstract polymer models

A new direct proof of convergence of cluster expansions for polymer (contour) models is given in an abstract setting. It does not rely on Kirkwood-Salsburg type equations or “combinatorics of trees.” A distinctive feature is that, at all steps, the considered clusters contain every polymer at most once.

A Rigorous Theory of Finite-Size Scaling at First-Order Phase Transitions

A large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered. The dependence of the finite-volume magnetizationMper(h, L) in cubes of sizeLdunder periodic boundary conditions on the external fieldh is analyzed. For the case where two phases coexist at the infinite-volume transition pointht, we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves likeMper(ht)+M tanh[MLd(h−ht)] withMper(h) denoting the infinite-volume magnetization and M=1/2[Mper(ht+0)−Mper(ht−0)]. Introducing the finite-size transition pointhm(L) as the point where the finite-volume susceptibility attains the maximum, we show that, in the case of asymmetric field-driven transitions, its shift isht−hm(L)=O(L−2d), in contrast to claims in the literature. Starting from the obvious observation that the number of stable phases has a local maximum at the transition point, we propose a new way of determining the pointhtfrom finite-size data with a shift that is exponentially small inL. Finally, the finite-size effects are discussed also in the case where more than two phases coexist.

Finite-size scaling for Potts models

Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd⩾2) undergoes a first-order phase transition as the inverse temperatureβ is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpΔE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted byΔβm(L)=(Inq/ΔE)L−d+O(L−2d) with respect to the infinite-volume transition pointβt. We also propose an alternative definition of the finite-volume transition temperatureβt(L) which might be useful for numerical calculations because it differs only by exponentially small corrections fromβt.

The Analysis of the Widom-Rowlinson Model by Stochastic Geometric Methods

We study the continuum Widom-Rowlinson model of interpenetrating spheres. Using a new geometric representation for this system we provide a simple percolation-based proof of the phase transition. We also use this representation to formulate the problem, and prove the existence of an interfacial tension between coexisting phases. Finally, we ascribe geometric (i.e. probabilistic) significance to the correlation functions which allows us to prove the existence of a sharp correlation length in the single-phase regime.

Low temperature phase diagrams for quantum perturbations of classical spin systems

AbstractWe consider a quantum spin system with Hamiltonian

Critical Region for Droplet Formation in the Two-Dimensional Ising Model

H = H^{(0)} + \lambda V,

Pathological Behavior of Renormalization-Group Maps at High Fields and Above the Transition Temperature

whereH(0) is diagonal in a basis ∣s〉=⊗x∣sx〉 which may be labeled by the configurationss={sx} of a suitable classical spin system on ℤd,