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.

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We consider classical lattice models describing first-order phase transitions, and study the finite-size scaling of the magnetization and susceptibility. In order to model the effects of an actual surface in systems like small magnetic clusters, we consider models with free boundary conditions. For a field driven transition with two coexisting phases at the infinite volume transition point h = h t , we prove that the low temperature finite volume magnetization m free (L, h) per site in a cubic volume of size L d behaves likewhere h χ (L) is the position of the maximum of the (finite volume) susceptibility and m ± are the infinite volume magnetizations at h = h t + 0 and h = h t − 0, respectively. We show that h χ (L) is shifted by an amound proportional to 1/L with respect to the infinite volume transitions point h t provided the surface free energies of the two phases at the transition point are different. This should be compared with the shift for periodic boundary conditons, which for an asymmetric transition with two coexisting phases is proportional only to 1/L 2d . One can consider also other definitions of finite volume transition points, as, for example, the position h U (L) of the maximum of the so called Binder cummulant U free (L, h). While it is again shifted by an amount proportional to 1/L with respect to the infinite volume transition point h t , its shift with respect to h χ (L) is of the much smaller order 1/L 2d . We give explicit formulas for the proportionality factors, and show that, in the leading 1/L 2d term, the relative shift is the same as that for periodic boundary conditions. † Heisenberg Fellow ‡ Partly supported by the grants GAČR 202/93/0499 and GAUK 376

PACS. 05.70.Fh -Phase transitions: general studies. PACS. 64.60.Cn -Order-disorder transformations, statistical mechanics of model systems. PACS. 75.10.Hk -Classical spin models.Abstract. -We consider liquid-vapor systems in finite volume V ⊂ R d at parameter values corresponding to phase coexistence and study droplet formation due to a fixed excess δN of particles above the ambient gas density. We identify a dimensionless parameter ∆ ∼ (δN ) (d+1)/d /V and a universal value ∆c = ∆c(d), and show that a droplet of the dense phase occurs whenever ∆ > ∆c, while, for ∆ < ∆c, the excess is entirely absorbed into the gaseous background. When the droplet first forms, it comprises a non-trivial, universal fraction of excess particles. Similar reasoning applies to generic two-phase systems at phase coexistence including solid/gas-where the "droplet" is crystalline-and polymorphic systems. A sketch of a rigorous proof for the 2D Ising lattice gas is presented; generalizations are discussed heuristically.Introduction. -The thermodynamics of droplets in systems with phase coexistence has been well understood since the pioneering works [1][2][3][4]. Recently, justifications of the classic results based on the first principles of statistical mechanics have been attempted-in both two [5][6][7][8][9] and higher [10-13] dimensions-and various thermodynamical predictions concerning macroscopic shapes have been rigorously established. However, the formation and dissolution of equilibrium droplets is among the less well-studied areas in statistical mechanics. Indeed, most of the aforementioned analysis has focused on the situation implicitly assumed in the classical derivations; namely, that the scale of the droplet is comparable with the scale of the system. As is known [7,8,[14][15][16]], this will not be the case when the parameter values are such that a droplet first forms. In this Letter, we underscore the region of the system parameters that is critical for the formation/dissolution of droplets. In particular, we isolate the mechanism by which the low-density phase copes with an excess of particles and pinpoint the critical amount of extra particles needed to cause a droplet to appear. Surprisingly, at the point of droplet formation, only a certain fraction of the excess goes into the droplet; the rest is absorbed by the bulk. Moreover, apart from a natural rescaling to dimensionless parameters, all of the above can be described in terms of universal quantities independent of the system particulars and the temperature.In the last few years, there has been some interest in questions related to droplet formation and dissolution with purported applicability in diverse areas such as nuclear fragmentation [17][18][19] and the stability of adatom islands on crystal surfaces [15,20]. Another issue, which is of practical significance in statistical mechanics, concerns the detection of first-order phase transitions by the study of small systems with fixed order parameter (magnetization) or fixed energy. Under these conditions, nonconv...

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