Geometric variational problems of statistical mechanics and of combinatorics

We present the geometric solutions of the various extremal problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibits the shape of a typical Young diagram and of a typical skyscraper.Comment: 10 page

Rigorous Non-Perturbative Ornstein-Zernike Theory for Ising Ferromagnets

We rigorously derive the Ornstein-Zernike asymptotics of the pair-correlation functions for finite-range Ising ferromagnets in any dimensions and at any temperature above critical

Pentaquark decay is suppressed by chirality conservation

It is shown, that if the pentaquark $\Theta^+ = uudd\bar{s}$ baryon can be represented by the local quark current $\eta_{\Theta}$, its decay $\Theta^+ \to n K^+ (p K^0)$ is forbidden in the limit of chirality conservation. The $\Theta^+$decay width $\Gamma$ is proportional to $\alpha^2_s < 0 | \bar{q} q | 0 >^2$, where $$, $q = u,d,s$ is quark condensate, and, therefore, is strongly suppressed. Also the polarization operator of the pentaquark current with isospin 1 is calculated using the operator product expansion and estimation for it mass is obtained .Comment: 4 pages, 1 fig, typos correcte

Random path representation and sharp correlations asymptotics at high-temperatures

We recently introduced a robust approach to the derivation of sharp asymptotic formula for correlation functions of statistical mechanics models in the high-temperature regime. We describe its application to the nonperturbative proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the proof, in the Ising case, to arbitrary odd-odd correlation functions. We discuss the fluctuations of connection paths (invariance principle), and relate the variance of the limiting process to the geometry of the equidecay profiles. Finally, we explain the relation between these results from Statistical Mechanics and their counterparts in Quantum Field Theory

Ornstein-Zernike Theory for the finite range Ising models above T_c

We derive precise Ornstein-Zernike asymptotic formula for the decay of the two-point function in the general context of finite range Ising type models on Z^d. The proof relies in an essential way on the a-priori knowledge of the strict exponential decay of the two-point function and, by the sharp characterization of phase transition due to Aizenman, Barsky and Fernandez, goes through in the whole of the high temperature region T > T_c. As a byproduct we obtain that for every T > T_c, the inverse correlation length is an analytic and strictly convex function of direction.Comment: 36 pages, 5 figure