We review the derivation of Wilson's differential equation in (infinitely) many variables, which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal “integration out” is performed. We show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. The differential equation approach gives a remarkable identity for “connected parts” and precise estimates which include criteria for convergence of iterated Mayer expansions. Applications include the Yukawa gas in two dimensions past theΒ=4π threshold and another derivation of some earlier results of Göpfert and Mack.
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J. Fröhlich, ed.,Scaling and Self Similarity in Physics—Renormalization in Statistical Mechanics and Dynamics (BirkhÄuser, Boston, 1983).
K. G. Wilson,Phys. Rev. B 4:3174 (1971); J. G. Kogut and K. G. Wilson,Phys. Rep. 12:263 (1974).
J. Polchinski, Renormalization and effective Lagrangians,Nucl. Phys. B 231:269 (1984).
G. Gallavotti, and F. Nicolo, Renormalization theory in four-dimensional scalar fields (I),Commun. Math. Phys. 100:545 (1985).
C. Newman, Unpublished work;J. Stat. Phys. 27:836 (1982).
J. Fröhlich and E. Seiler,Helv. Phys. Acta 49:889 (1976).
D. C. Brydges, A short course on cluster expansions, Appendix A, inCritical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds. (Elsevier, 1986).
G. Battle, and P. Federbush, A note on cluster expansions, tree graph identities, extra 1/N! factors!Lett. Math. Phys. 8:55 (1984).
D. C. Brydges, Convergence of Mayer expansions,J. Stat. Phys. 42:425 (1984).
J. Fröhlich,Commun. Math. Phys. 47:233 (1976).
G. Benfatto, An iterated Mayer expansion for the Yukawa gas,J. Stat. Phys. 41:671 (1985).
M. Göpfert and G. Mack,Commun. Math. Phys. 81:97 (1981);82:545 (1982).
G. Gallavotti, Renormalization theory and ultra-violet stability for scalar fields via renormalisation group techniques,Rev. Mod. Phys. 57:471 (1985).
G. Benfatto, G. Gallavotti, and F. Nicolo, On the massive sine Gordon equation in the first few regions of collapse,Commun. Math. Phys. 83:387 (1982).
F. Nicolo, J. Benn, and A. Steinman, On the massive sine Gordon equation in all regions of collapse,Commun. Math. Phys. 105:291 (1986).
G. Benfatto, G. Gallavotti, and F. Nicolo, In preparation.
G. Gallavotti and F. Nicolo, The screening phase transition in the two dimensional Coulomb gas,J. Stat. Phys. 39:133 (1985).
J. Imbrie, Iterated Mayer expansions and their applications to Coulomb systems, inScaling and Self-Similanty in Physics—Renormalization in Statistical Mechanics and Dynamics, J. Frölich, ed. (BirkhÄuser, Boston, 1983).
V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978), p. 256.
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Brydges, D.C., Kennedy, T. Mayer expansions and the Hamilton-Jacobi equation. J Stat Phys 48, 19–49 (1987). https://doi.org/10.1007/BF01010398