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Critical correlation functions for the 4-dimensional weakly self-avoiding walk and $$n$$-component $$|\varphi|^4$$ model. (English) Zbl 1342.82070
The authors study the critical behavior of the continuous-time weakly self-avoiding walk and of the $$n$$-component $$|\varphi|^4$$ model, for all $$n\geq 1$$, in the upper critical dimension $$d=4$$. Using a rigorous renormalization group method the corresponding critical correlation functions are studied. It is proven that, for the $$|\varphi|^4$$ model the critical two-point function has $$|x|^{-2}$$ (Gaussian) decay asymptotically, for $$n\geq 1$$. Moreover the authors determine the asymptotic decay of the critical correlations of the squares of components of $$\varphi$$, including the logarithmic corrections to Gaussian scaling, for $$n\geq 1$$. For the continuous-time weakly self-avoiding walk, the authors determine the decay of the critical generating function for the “watermelon” network consisting of $$p$$ weakly mutually-and self-avoiding walks, for all $$p\geq 1$$, including the logarithmic corrections. For both models, the approach to the critical point is studied and the existence of logarithmic corrections to scaling for certain correlation functions is proved.

##### MSC:
 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B28 Renormalization group methods in equilibrium statistical mechanics
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##### References:
 [1] Abdesselam, A., Chandra, A., Guadagni, G.: Rigorous quantum field theory functional integrals over the $$p$$-adics I: anomalous dimensions (2013). arXiv:1302.5971 [2] Aharony, A.; Domb, C. (ed.); Green, M.S. (ed.), Dependence of universal critical behaviour on symmetry and range of interaction, 357-424, (1976), London [3] Ahlers, G.; Kornblit, A.; Guggenheim, H.J., Logarithmic corrections to the Landau specific heat near the Curie temperature of the dipolar Ising ferromagnet litbf_{4}, Phys. Rev. Lett., 34, 1227-1230, (1975) [4] Aizenman, M., Geometric analysis of $${φ^{4}}$$ fields and Ising models, parts I and II, Commun. Math. Phys., 86, 1-48, (1982) · Zbl 0533.58034 [5] Aizenman, M., The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory, Commun. Math. Phys., 97, 91-110, (1985) · Zbl 0573.60076 [6] Aizenman, M.; Duminil-Copin, H.; Sidoravicius, V., Random currents and continuity of Ising model’s spontaneous magnetization, Commun. Math. Phys., 334, 719-742, (2015) · Zbl 1315.82004 [7] Aizenman, M.; Fernández, R., On the critical behavior of the magnetization in high dimensional Ising models, J. Stat. Phys., 44, 393-454, (1986) · Zbl 0629.60106 [8] Aizenman, M.; Graham, R., On the renormalized coupling constant and the susceptibility in $${ϕ_{4}^{4}}$$ field theory and the Ising model in four dimensions, Nucl. Phys. B, 225, 261-288, (1983) [9] Aragão de Carvalho, C.; Caracciolo, S.; Fröhlich, J., Polymers and $${g|ϕ|^{4}}$$ theory in four dimensions, Nucl. Phys. B, 215, 209-248, (1983) [10] Bałaban, T.: A low temperature expansion and “spin wave picture” for classical $$N$$-vector models. In: Rivasseau, V. (ed.) Constructive Physics Results in Field Theory, Statistical Mechanics and Condensed Matter Physics. Lecture Notes in Physics, vol. 446. Springer, Berlin (1995) [11] Bałaban, T.; O’Carroll, M., Low temperature properties for correlation functions in classical $$N$$-vector spin models, Commun. Math. Phys., 199, 493-520, (1999) · Zbl 0932.81024 [12] Bauerschmidt, R., A simple method for finite range decomposition of quadratic forms and Gaussian fields, Probab. Theory Relat. Fields, 157, 817-845, (2013) · Zbl 1347.60037 [13] Bauerschmidt, R.; Brydges, D.C.; Slade, G., Scaling limits and critical behaviour of the 4-dimensional $$n$$-component $${|φ|^{4}}$$ spin model, J. Stat. Phys, 157, 692-742, (2014) · Zbl 1308.82026 [14] Bauerschmidt, R.; Brydges, D.C.; Slade, G., Critical two-point function of the 4-dimensional weakly self-avoiding walk, Commun. Math. Phys., 338, 169-193, (2015) · Zbl 1320.82031 [15] Bauerschmidt, R.; Brydges, D.C.; Slade, G., Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis, Commun. Math. Phys., 337, 817-877, (2015) · Zbl 1318.60049 [16] Bauerschmidt, R.; Brydges, D.C.; Slade, G., A renormalisation group method. III. perturbative analysis, J. Stat. Phys, 159, 492-529, (2015) · Zbl 1319.82008 [17] Bauerschmidt, R.; Brydges, D.C.; Slade, G., Structural stability of a dynamical system near a non-hyperbolic fixed point, Ann. Henri Poincaré, 16, 1033-1065, (2015) · Zbl 1347.37041 [18] Bovier, A.; Felder, G.; Fröhlich, J., On the critical properties of the edwards and the self-avoiding walk model of polymer chains, Nucl. Phys. B, 230, 119-147, (1984) [19] Brézin, E.; Le Guillou, J.C.; Zinn-Justin, J., Approach to scaling in renormalized perturbation theory, Phys. Rev. D, 8, 2418-2430, (1973) [20] Brydges, D.; Evans, S.N.; Imbrie, J.Z., Self-avoiding walk on a hierarchical lattice in four dimensions, Ann. Probab., 20, 82-124, (1992) · Zbl 0742.60067 [21] Brydges, D.C., Dahlqvist, A., Slade, G.: The strong interaction limit of continuous-time weakly self-avoiding walk. In: Deuschel, J.-D., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds.) Probability in Complex Physical Systems: In Honour of Erwin Bolthausen and Jürgen Gärtner. Proceedings in Mathematics, vol. 11, pp. 275-287. Springer, Berlin (2012) · Zbl 1246.82042 [22] Brydges, D.C.; Fröhlich, J.; Spencer, T., The random walk representation of classical spin systems and correlation inequalities, Commun. Math. Phys., 83, 123-150, (1982) [23] Brydges, D.C.; Guadagni, G.; Mitter, P.K., Finite range decomposition of Gaussian processes, J. Stat. Phys., 115, 415-449, (2004) · Zbl 1157.82304 [24] Brydges, D.C.; Imbrie, J.Z., Branched polymers and dimensional reduction, Ann. Math., 158, 1019-1039, (2003) · Zbl 1140.82314 [25] Brydges, D.C.; Imbrie, J.Z., End-to-end distance from the green’s function for a hierarchical self-avoiding walk in four dimensions, Commun. Math. Phys., 239, 523-547, (2003) · Zbl 1036.82011 [26] Brydges, D.C.; Imbrie, J.Z., Green’s function for a hierarchical self-avoiding walk in four dimensions, Commun. Math. Phys., 239, 549-584, (2003) · Zbl 1087.82010 [27] Brydges, D.C.; Imbrie, J.Z.; Slade, G., Functional integral representations for self-avoiding walk, Probab. Surv., 6, 34-61, (2009) · Zbl 1193.82014 [28] Brydges, D.C.; Slade, G., A renormalisation group method. I. Gaussian integration and normed algebras, J. Stat. Phys., 159, 421-460, (2015) · Zbl 1317.82013 [29] Brydges, D.C.; Slade, G., A renormalisation group method. II. approximation by local polynomials, J. Stat. Phys., 159, 461-491, (2015) · Zbl 1317.82014 [30] Brydges, D.C.; Slade, G., A renormalisation group method. IV. stability analysis, J. Stat. Phys., 159, 530-588, (2015) · Zbl 1317.82015 [31] Brydges, D.C.; Slade, G., A renormalisation group method. V. A single renormalisation group step, J. Stat. Phys., 159, 589-667, (2015) · Zbl 1317.82016 [32] Cardy J.: Scaling and Renormalization in Statistical Physics. Cambridge University Press, Cambridge (1996) · Zbl 0914.60002 [33] Chelkak, D.; Smirnov, S., Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., 189, 515-580, (2012) · Zbl 1257.82020 [34] Dimock, J.; Hurd, T.R., A renormalization group analysis of correlation functions for the dipole gas, J. Stat. Phys., 66, 1277-1318, (1992) · Zbl 0925.82083 [35] Dunlop, F.; Newman, C.M., Multicomponent field theories and classical rotators, Commun. Math. Phys., 44, 223-235, (1975) [36] Duplantier, B., Polymer chains in four dimensions, Nucl. Phys. B, 275, 319-355, (1986) [37] Duplantier, B., Intersections of random walks. A direct renormalization approach, Commun. Math. Phys., 117, 279-329, (1988) · Zbl 0652.60073 [38] Duplantier, B., Statistical mechanics of polymer networks of any topology, J. Stat. Phys., 54, 581-680, (1989) [39] Dynkin, E.B., Markov processes as a tool in field theory, J. Funct. Anal., 50, 167-187, (1983) · Zbl 0522.60078 [40] Falco, P.: Critical exponents of the two dimensional Coulomb gas at the Berezinskii-Kosterlitz-Thouless transition (2013). arXiv:1311.2237 · Zbl 1254.82012 [41] Felder, G.; Fröhlich, J., Intersection probabilities of simple random walks: a renormalization group approach, Commun. Math. Phys., 97, 111-124, (1985) · Zbl 0573.60065 [42] Feldman, J.; Magnen, J.; Rivasseau, V.; Sénéor, R., Construction and Borel summability of infrared $${Φ^{4}_{4}}$$ by a phase space expansion, Commun. Math. Phys., 109, 437-480, (1987) [43] Fernández R., Fröhlich J., Sokal A.D.: Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer, Berlin (1992) · Zbl 0761.60061 [44] Fröhlich, J., On the triviality of $${φ_{d}^{4}}$$ theories and the approach to the critical point in $${d ≥ 4}$$ dimensions, Nucl. Phys. B, 200, 281-296, (1982) [45] Gawȩdzki, K.; Kupiainen, A., Massless lattice $${φ^{4}_{4}}$$ theory: rigorous control of a renormalizable asymptotically free model, Commun. Math. Phys., 99, 199-252, (1985) [46] Gawȩdzki, K., Kupiainen, A.: Asymptotic freedom beyond perturbation theory. In: Osterwalder, K., Stora, R., (eds.) Critical Phenomena, Random Systems, Gauge Theories, Amsterdam, (1986). [North-Holland. Les Houches (1984)] [47] de Gennes, P.G., Exponents for the excluded volume problem as derived by the Wilson method, Phys. Lett. A, 38, 339-340, (1972) [48] Glimm J., Jaffe A.: Quantum Physics, A Functional Integral Point of View, 2nd edn. Springer, Berlin (1987) · Zbl 0461.46051 [49] Grimmett, G.R.; Manolescu, I., Bond percolation on isoradial graphs: criticality and universality, Probab. Theory Relat. Fields, 159, 273-327, (2014) · Zbl 1296.60263 [50] Hara, T., A rigorous control of logarithmic corrections in four dimensional $${φ^{4}}$$ spin systems. I. trajectory of effective Hamiltonians, J. Stat. Phys., 47, 57-98, (1987) [51] Hara, T.; Hattori, T.; Watanabe, H., Trivitality of hierarchical Ising model in four dimensions, Commun. Math. Phys., 220, 13-40, (2001) · Zbl 1001.82044 [52] Hara, T.; Slade, G., Self-avoiding walk in five or more dimensions. I. the critical behaviour, Commun. Math. Phys., 147, 101-136, (1992) · Zbl 0755.60053 [53] Hara, T.; Tasaki, H., A rigorous control of logarithmic corrections in four dimensional $${φ^{4}}$$ spin systems. II. critical behaviour of susceptibility and correlation length, J. Stat. Phys., 47, 99-121, (1987) [54] Holmes, M.; Járai, A.A.; Sakai, A.; Slade, G., High-dimensional graphical networks of self-avoiding walks, Canad. J. Math., 56, 77-114, (2004) · Zbl 1137.82313 [55] Iagolnitzer, D.; Magnen, J., Polymers in a weak random potential in dimension four: rigorous renormalization group analysis, Commun. Math. Phys., 162, 85-121, (1994) · Zbl 0796.60105 [56] Kenyon, R.; Winkler, P., Branched polymers, Am. Math. Mon., 116, 612-628, (2009) · Zbl 1229.82172 [57] Larkin, A.I., Khmel’Nitskiĭ, D.E.: Phase transition in uniaxial ferroelectrics. Sov. Phys. JETP 29, 1123-1128 (1969). [English translation of Zh. Eksp. Teor. Fiz. 56, 2087-2098 (1969)] · Zbl 0985.60090 [58] Lawler, G.F., Intersections of random walks in four dimensions. II, Commun. Math. Phys., 97, 583-594, (1985) · Zbl 0585.60069 [59] Lawler G.F.: Intersections of Random Walks. Birkhäuser, Boston (1991) · Zbl 1228.60004 [60] Lawler, G.F., Escape probabilities for slowly recurrent sets, Probab. Theory Relat. Fields, 94, 91-117, (1992) · Zbl 0767.60062 [61] Le Jan, Y., On the Fock space representation of functionals of the occupation field and their renormalization, J. Funct. Anal., 80, 88-108, (1988) · Zbl 0671.60064 [62] Madras N., Slade G.: The Self-Avoiding Walk. Birkhäuser, Boston (1993) · Zbl 0780.60103 [63] McKane, A.J., Reformulation of $${n → 0}$$ models using anticommuting scalar fields, Phys. Lett. A, 76, 22-24, (1980) [64] Mitter, P.K.; Scoppola, B., The global renormalization group trajectory in a critical supersymmetric field theory on the lattice $${{{\mathbb{Z}}}^{3}}$$, J. Stat. Phys., 133, 921-1011, (2008) · Zbl 1161.82310 [65] Parisi, G.; Sourlas, N., Self-avoiding walk and supersymmetry, J. Phys. Lett., 41, l403-l406, (1980) [66] Park, Y.M., Direct estimates on intersection probabilities of random walks, J. Stat. Phys., 57, 319-331, (1989) · Zbl 0716.60081 [67] Sakai, A., Lace expansion for the Ising model, Commun. Math. Phys., 272, 283-344, (2007) · Zbl 1133.82007 [68] Sakai, A., Application of the lace expansion to the $${φ^4}$$ model, Commun. Math. Phys., 336, 619-648, (2015) · Zbl 1315.82006 [69] Simon, B.; Griffiths, R.B., The $${(ϕ^{4})_{2}}$$ field theory as a classical Ising model, Commun. Math. Phys., 33, 145-164, (1973) [70] Slade, G.: The Lace Expansion and its Applications. Springer, Berlin (2006). (Lecture Notes in Mathematics, vol. 1879. Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004) · Zbl 1113.60005 [71] Smirnov, S., Critical percolation in the plane: conformal invariance, cardy’s formula, scaling limits, C. R. Math. Acad. Sci. Paris, 333, 239-244, (2001) · Zbl 0985.60090 [72] Smirnov, S.: Critical percolation in the plane. I. Conformal invariance and Cardy’s formula. II. Continuum scaling limit (2001). arXiv:0909.4499 · Zbl 0985.60090 [73] Smirnov, S.; Werner, W., Critical exponents for two-dimensional percolation, Math. Res. Lett., 8, 729-744, (2001) · Zbl 1009.60087 [74] Symanzik, K.: Euclidean quantum field theory. In: Jost, R. Local Quantum Field Theory, Academic Press, New York (1969) [75] Wegner, F.J.; Riedel, E.K., Logarithmic corrections to the molecular-field behavior of critical and tricritical systems, Phys. Rev. B, 7, 248-256, (1973) [76] Wilson, K.G.; Kogut, J., The renormalization group and the ε expansion, Phys. Rep., 12, 75-200, (1974)
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