Summary
We consider simple random walk onZ d perturbed by a factor exp[βT −P J T], whereT is the length of the walk and\(J_T = \sum\nolimits_{0 \leqslant i< j \leqslant T} \delta _{\omega (i),\omega (j)} \). Forp=1 and dimensionsd≥2, we prove that this walk behaves diffusively for all − ∞ < β <0, with β0 > 0. Ford>2 the diffusion constant is equal to 1, but ford=2 it is renormalized. Ford=1 andp=3/2, we prove diffusion for all real β (positive or negative). Ford>2 the scaling limit is Brownian motion, but ford≤2 it is the Edwards model (with the “wrong” sign of the coupling when β>0) which governs the limiting behaviour; the latter arises since for\(p = \frac{{4 - d}}{2}\),T −p J T is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.
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References
S. Albeverio and X.Y. Zhou. A modified Domb-Joyce model in four dimensions. Preprint, (1993).
P. Billingsley,Convergence of Probability Measures, John Wiley and Sons, New York (1968).
E. Bolthausen. On the construction of the three dimensional polymer measure.Probab. Theory Relat. Fields,97:81–101, (1993).
E. Bolthausen. Localization of a two-dimensional random walk with an attractive path interaction.Ann. Probab.,22:875–918, (1994).
E. Bolthausen and U. Schmock. On self-attractingd-dimensional random walks. Preprint, (1994).
E. Bolthausen and U. Schmock. On self-attracting random walks. In M.C. Cranston and M.A. Pinsky, editors,Stochastic Analysis, Providence, (1995), American Mathematical Society. Proceedings of Symposia in Pure Mathematics, Volume 57.
A.N. Borodin. On the asymptotic behavior of local times of recurrent random wllks with finite variance.Theory Probab. Appl.,26:758–772, (1981).
A.N. Borodin. Brownian local time.Russian Math. Surveys,44:1–51, (1989).
A. Bovier, G. Felder, and J. Fröhlich. On the critical properties of the Edwards and the self-avoiding walk model of polymer chains.Nucl. Phys. B,230 [FS10]:119–147, (1984).
R. Brak, A.J. Guttmann, and S.G. Whittington. A collapse transition in a directed walk model.J. Phys. A: Math. Gen.,25:2437–2446, (1992).
R. Brak, A.L. Owezarek, and T. Prellberg. A scaling theory of the collapse transition in geometric cluster models of polymers and vesicles.J. Phys. A: Math. Gen.,26:4565–4579, (1993).
D. Brydges, S.N. Evans, and I.Z. Imbrie. Self-avoiding walk on a hierarchical lattice in four dimensions.Ann. Probab.,20:82–124, (1992).
D.C. Brydges and G. Slade. A collapse transition for self-attracting walks.Resenhas do Instituto da Matemática e Estatística da Universidade de São Paulo,1 363–372, (1994).
S. Caracciolo, G. Parisi, and A. Pelissetto. Random walks with short-range interaction and mean-field behavior.J. Stat. Phys.,77:519–543, (1994).
J. Fröhlich and Y.M. Park. Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems.Commun. Math. Phys.,59:235–266, (1978).
A. Greven and F. den Hollander. A variational characterization of the speed of a one-dimensional self-repellent random walk.Ann. Appl. Probab.,3:1067–1099, (1993).
D. Iagolnitzer and J. Magnen. Polymers in a weak random potential in dimension four: rigorous renormalization group analysis.Commun. Math. Phys.,162:85–121, (1994).
T. Kennedy. Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty.J. Stat. Phys.,77:565–579, (1994).
G.F. Lawler.Intersections of Random Walks. Birkhäuser, Boston, (1991).
J.-F. Le Gall. Sur le temps local d'intersection du mouvement brownien plan et la methode de renormalization de Varadhan. In J. Azéma and M. Yor, editors.Séminaire de Probabilités XIX.Lecture Notes in Mathematics #1123. Berlin, (1985), Springer.
J.-F. Le Gall. Propriétés d'intersection des marches aléatoires I. Convergence vers le temps local d'intersection.Commun. Math. Phys.,104:471–507, (1986).
J.-F. Le Gall. Exponential moments for the renormalized self-intersection local time of planar Brownian motion. In J. Azéma, P.A. Meyer, and M. Yor, editors,Séminaire de Probabilités XXVIII.Lecture Notes in Mathematics #1583, Berlin, (1994). Springer.
J.L. Lebowitz, H.A. Rose, and E.R. Speer. Statistical mechanics of the nonlinear Schrödinger equation.J. Stat. Phys.,50:657–687, (1988).
Y. Oono, On the divergence of the perturbation series for the excluded-volume problem in polymers.J. Phys. Soc. Japan,39:25–29, (1975).
Y. Oono. On the divergence of the perturbation series for the excluded-volume problem in polymers. II. Collapse of a single chain in poor solvents.J. Phys. Soc. Japan,41:787–793, (1976).
E. Perkins. Weak invariance principles for local time.Z. Wahrsch. verw. Gebiete,60:437–451, (1982).
J. Rosen, Self-intersections of random fields.Ann. Probab.,12:108–119, (1984).
J. Rosen. Random walks and intersection local time.Ann. Probab.,18:959–977, (1990).
U. Schmock. Convergence of the normalized one-dimensional Wiener sausage path measures to a mixture of Brownian taboo processes.Stochastics and Stochastic Reports,29:171–183, (1989).
F. Spitzer,Principles of Random Walk. Springer, New York, 2nd edition, (1976).
A. Stoll. Invariance principles for Brownian intersection local time and polymer measures.Math. Scand.,64:133–160, (1989).
A.-S. Sznitman. On the confinement property of two-dimensional Brownian motion among Poissonian obstacles.Commun. Pure Appl. Math.,44:1137–1170, (1991).
S.R.S. Varadhan. Appendix to: Euclidean quantum field theory, by K. Symanzik. In R. Jost, editor,Local Quantum Field Theory. New York, (1969). Academic Press.
J. Westwater. On Edwards' model for long polymer chains.Commun. Math. Phys.,72:131–174, (1980).
J. Westwater. On Edwards' model for long polymer chains III. Borel summability.Commun. Math. Phys.,84:459–470, (1982).
J. Westwater. On Edwards' model for long polymer chains. In S. Albeverio and P. Blanchard, editors,Trends and Developments in the Eighties. Bielefeld Encounters in Mathematical Physics IV/V. World Scientific, Singapore, (1985).
H. Zoladek. One-dimensional random walk with self-interaction.J. Stat. Phys.,47:543–550, (1987).