Abstract
We consider the critical behaviour of long-range \({O(n)}\) models (\({n \ge 0}\)) on \({\mathbb{Z}^d}\), with interaction that decays with distance r as \({r^{-(d+\alpha)}}\), for \({\alpha \in (0,2)}\). For \({n \ge 1}\), we study the n-component \({|\varphi|^4}\) lattice spin model. For \({n =0}\), we study the weakly self-avoiding walk via an exact representation as a supersymmetric spin model. These models have upper critical dimension \({d_c=2\alpha}\). For dimensions \({d=1,2,3}\) and small \({\epsilon > 0}\), we choose \({\alpha = \frac 12 (d+\epsilon)}\), so that \({d=d_c-\epsilon}\) is below the upper critical dimension. For small \({\epsilon}\) and weak coupling, to order \({\epsilon}\) we prove the existence of and compute the values of the critical exponent \({\gamma}\) for the susceptibility (for \({n \ge 0}\)) and the critical exponent \({\alpha_H}\) for the specific heat (for \({n \ge 1}\)). For the susceptibility, \({\gamma = 1 + \frac{n+2}{n+8} \frac \epsilon\alpha + O(\epsilon^2)}\), and a similar result is proved for the specific heat. Expansion in \({\epsilon}\) for such long-range models was first carried out in the physics literature in 1972. Our proof adapts and applies a rigorous renormalisation group method developed in previous papers with Bauerschmidt and Brydges for the nearest-neighbour models in the critical dimension \({d=4}\), and is based on the construction of a non-Gaussian renormalisation group fixed point. Some aspects of the method simplify below the upper critical dimension, while some require different treatment, and new ideas and techniques with potential future application are introduced.
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References
Abdesselam A.: A complete renormalization group trajectory between two fixed points. Commun. Math. Phys. 276, 727–772 (2007)
Abdesselam, A.: Towards three-dimensional conformal probability. arXiv:1511.03180
Abdesselam, A., Chandra, A., Guadagni, G.: Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions. arXiv:1302.5971
Adams, S., Kotecký, R., Müller, S.: Strict convexity of the surface tension for non-convex potentials. arXiv:1606.09541
Aizenman M.: Geometric analysis of \({\varphi^4}\) fields and Ising models, parts I and II. Commun. Math. Phys. 86, 1–48 (1982)
Aizenman M., Fernández R.: Critical exponents for long-range interactions. Lett. Math. Phys. 16, 39–49 (1988)
Amit D.J.: Field Theory, the Renormalization Group, and Critical Phenomena, 2nd edn. World Scientific, Singapore (1984)
Balaban 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. Springer Lecture Notes in Physics, vol. 446, Springer, Berlin (1995)
Balaban T., Feldman J., Knörrer H., Trubowitz E.: Complex Bosonic many-body models: overview of the small field parabolic flow. Ann. Henri Poincaré 18, 2873–2903 (2017)
Balaban T., O’Carroll M.: Low temperature properties for correlation functions in classical N-vector spin models. Commun. Math. Phys. 199, 493–520 (1999)
Bauerschmidt R.: A simple method for finite range decomposition of quadratic forms and Gaussian fields. Probab. Theory Relat. Fields 157, 817–845 (2013)
Bauerschmidt, R., Brydges, D.C., Slade, G.: Introduction to a renormalisation group method for critical phenomena. https://www.math.ubc.ca/~slade/brief-frd.pdf (in preparation)
Bauerschmidt R., Brydges D.C., Slade G.: Scaling limits and critical behaviour of the 4-dimensional n-component \({|\varphi|^4}\) spin model. J. Stat. Phys. 157, 692–742 (2014)
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)
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)
Bauerschmidt R., Brydges D.C., Slade G.: A renormalisation group method. III. Perturbative analysis. J. Stat. Phys. 159, 492–529 (2015)
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)
Bauerschmidt R., Slade G., Tomberg A., Wallace B.C.: Finite-order correlation length for 4-dimensional weakly self-avoiding walk and \({|\varphi|^4}\) spins. Ann. Henri Poincaré 18, 375–402 (2017)
Behan C., Rastelli L., Rychkov S., Zan B.: A scaling theory for long-range to short-range crossover and an infrared duality. J. Phys. A Math. Theor. 50, 354002 (2017)
Bendikov A., Cygan W.: \({\alpha}\)-stable random walk has massive thorns. Colloq. Math. 138, 105–129 (2015)
Bendikov, A., Cygan, W., Trojan, B.: Limit theorems for random walks. Stoch. Proc. Appl. 127, 3268–3290 (2017)
Blumenthal R.M., Getoor R.K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960)
Brezin E., Parisi G., Ricci-Tersenghi F.: The crossover region between long-range and short-range interactions for the critical exponents. J. Stat. Phys. 157, 855–868 (2014)
Brydges D., Dimock J., Hurd T.R.: A non-Gaussian fixed point for \({\phi^4}\) in \({4-\epsilon}\) dimensions. Commun. Math. Phys. 198, 111–156 (1998)
Brydges D., Evans S.N., Imbrie J.Z.: Self-avoiding walk on a hierarchical lattice in four dimensions. Ann. Probab. 20, 82–124 (1992)
Brydges D., Mitter P.K.: On the convergence to the continuum of finite range lattice covariances. J. Stat. Phys. 147, 716–727 (2012)
Brydges D.C.: Lectures on the renormalisation group. In: Sheffield, S., Spencer, T. (eds.) Statistical Mechanics. IAS/Park City Mathematics Series, vol. 16, pp. 7–93. American Mathematical Society, Providence (2009)
Brydges D.C., Guadagni G., Mitter P.K.: Finite range decomposition of Gaussian processes. J. Stat. Phys. 115, 415–449 (2004)
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)
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)
Brydges D.C., Imbrie J.Z., Slade G.: Functional integral representations for self-avoiding walk. Probab. Surv. 6, 34–61 (2009)
Brydges D.C., Mitter P.K., Scoppola B.: Critical \({({\Phi}^4)_{3,\epsilon}}\). Commun. Math. Phys. 240, 281–327 (2003)
Brydges D.C., Muñoz Maya I.: An application of Berezin integration to large deviations. J. Theor. Probab. 4, 371–389 (1991)
Brydges D.C., Slade G.: A renormalisation group method. I. Gaussian integration and normed algebras. J. Stat. Phys. 159, 421–460 (2015)
Brydges D.C., Slade G.: A renormalisation group method. II. Approximation by local polynomials. J. Stat. Phys. 159, 461–491 (2015)
Brydges D.C., Slade G.: A renormalisation group method. IV. Stability analysis. J. Stat. Phys. 159, 530–588 (2015)
Brydges D.C., Slade G.: A renormalisation group method. V. A single renormalisation group step. J. Stat. Phys. 159, 589–667 (2015)
Brydges D.C., Spencer T.: Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys. 97, 125–148 (1985)
Chen L.-C., Sakai A.: Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Relat. Fields 142, 151–188 (2008)
Chen L.-C., Sakai A.: Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation. Ann. Probab. 39, 507–548 (2011)
Chen L.-C., Sakai A.: Critical two-point functions for long-range statistical-mechanical models in high dimensions. Ann. Probab. 43, 639–681 (2015)
Collet P., Eckmann J.-P.: A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics. Lecture Notes in Physics, vol. 74. Springer, Berlin (1978)
Falco, P.: Critical exponents of the two dimensional Coulomb gas at the Berezinskii–Kosterlitz–Thouless transition. arXiv:1311.2237
Feldman J., Magnen J., Rivasseau V., Sénéor R.: Construction and Borel summability of infrared \({\Phi^4_4}\) by a phase space expansion. Commun. Math. Phys. 109, 437–480 (1987)
Fernández R., Fröhlich J., Sokal A.D.: Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer, Berlin (1992)
Fisher M.E., Ma S., Nickel B.G.: Critical exponents for long-range interactions. Phys. Rev. Lett. 29, 917–920 (1972)
Fröhlich J.: On the triviality of \({\varphi_d^4}\) theories and the approach to the critical point in \({d \geq 4}\) dimensions. Nucl. Phys. B 200([FS4]), 281–296 (1982)
Gawȩdzki K., Kupiainen A.: Non-Gaussian fixed points of the block spin transformation. Hierarchical model approximation. Commun. Math. Phys. 89, 191–220 (1983)
Gawȩdzki K., Kupiainen A.: Non-Gaussian scaling limits: Hierarchical model approximation. J. Stat. Phys. 35, 267–284 (1984)
Gawȩdzki K., Kupiainen A.: Massless lattice \({\varphi^4_4}\) theory: rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys. 99, 199–252 (1985)
Gawȩdzki K., Kupiainen A.: Asymptotic freedom beyond perturbation theory. In: Osterwalder, K., Stora, R. (eds) Critical Phenomena, Random Systems, Gauge Theories, North-Holland, Amsterdam (1986)
Giuliani A., Mastropietro V., Toninelli F.L.: Height fluctuations in interacting dimers. Ann. I. Henri Poincaré Probab. Stat. 53, 98–168 (2017)
Glimm J., Jaffe A.: Quantum Physics, A Functional Integral Point of View, 2nd edn. Springer, Berlin (1987)
Grigor’yan A., Telcs A.: Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324, 521–556 (2002)
Guida R., Zinn-Justin J.: Critical exponents of the N-vector model. J. Phys. A Math. Gen. 31, 8103–8121 (1998)
Hairer, M.: Regularity structures and the dynamical \({{\Phi}^4_3}\) model. arXiv:1508.05261
Hara T.: A rigorous control of logarithmic corrections in four dimensional \({\varphi^4}\) spin systems. I. Trajectory of effective Hamiltonians. J. Stat. Phys. 47, 57–98 (1987)
Hara T., Hattori T., Watanabe H.: Triviality of hierarchical Ising model in four dimensions. Commun. Math. Phys. 220, 13–40 (2001)
Hara T., Slade G.: Self-avoiding walk in five or more dimensions. I. The critical behaviour. Commun. Math. Phys. 147, 101–136 (1992)
Hara T., Tasaki H.: A rigorous control of logarithmic corrections in four dimensional \({\varphi^4}\) spin systems. II. Critical behaviour of susceptibility and correlation length. J. Stat. Phys. 47, 99–121 (1987)
Heydenreich M.: Long-range self-avoiding walk converges to alpha-stable processes. Ann. I. Henri Poincaré Probab. Stat. 47, 20–42 (2011)
Heydenreich M., van der Hofstad R., Sakai A.: Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk. J. Stat. Phys. 132, 1001–1049 (2008)
Honkonen J., Yu M. Nalimov: Crossover between field theories with short-range and long-range exchange or correlations. J. Phys. A Math. Gen. 22, 751–763 (1989)
Iagolnitzer D., Magnen J.: Polymers in a weak random potential in dimension four: rigorous renormalization group analysis. Commun. Math. Phys. 162, 85–121 (1994)
Kato T.: Note on fractional powers of linear operators. Proc. Jpn. Acad. 36, 94–96 (1960)
Kompaniets M.V., Panzer E.: Minimally subtracted six loop renormalization of \({{O}(n)}\)-symmetric \({\phi^4}\) theory and critical exponents. Phys.Rev. D 96, 036016 (2017)
Kupiainen A.: Renormalization group and stochastic PDE’s. Ann. Henri Poincaré 17, 497–535 (2016)
Lawler G.F.: Intersections of Random Walks. Birkhäuser, Boston (1991)
Le Guillou J.C., Zinn-Justin J.: Accurate critical exponents from the \({\varepsilon}\)-expansion. J. Phys. Lett. 46, L137–L141 (1985)
Lodhia A., Sheffield S., Sun X., Watson S.S.: Fractional Gaussian fields: a survey. Probab. Surv. 13, 1–56 (2016)
Lohmann, M., Slade, G., Wallace, B.C.: Critical two-point function for long-range \({O(n)}\) models below the upper critical dimension. J. Stat. Phys. arXiv:1705.08540 (in press)
Loomis L.H., Sternberg S.: Advanced Calculus. World Scientific, Singapore (2014)
Luttinger J.M.: The asymptotic evaluation of a class of path integrals. II. J. Math. Phys. 24, 2070–2073 (1983)
Madras N., Slade G.: The Self-Avoiding Walk. Birkhäuser, Boston (1993)
McKane A.J.: Reformulation of \({n \to 0}\) models using anticommuting scalar fields. Phys. Lett. A 76, 22–24 (1980)
Mitter, P.: Long range ferromagnets: renormalization group analysis. https://hal.archives-ouvertes.fr/cel-01239463 (2013)
Mitter, P.K.: On a finite range decomposition of the resolvent of a fractional power of the Laplacian. J. Stat. Phys. 163, 1235–1246 (2016). Erratum: J. Stat. Phys. 166, 453–455 (2017)
Mitter P.K.: On a finite range decomposition of the resolvent of a fractional power of the Laplacian II. The torus. J. Stat. Phys. 168, 986–999 (2017)
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)
Norris J.R.: Markov Chains. Cambridge University Press, Cambridge (1997)
Parisi G., Sourlas N.: Self-avoiding walk and supersymmetry. J. Phys. Lett. 41, L403–L406 (1980)
Paulos M.F., Rychkov S., van Rees B.C., Zan B.: Conformal invariance in the long-range Ising model. Nucl. Phys. B 902, 246–291 (2016)
Sak J.: Recursion relations and fixed points for ferromagnets with long-range interactions. Phys. Rev. B 8, 281–285 (1973)
Sakai A.: Application of the lace expansion to the \({\varphi^4}\) model. Commun. Math. Phys. 336, 619–648 (2015)
Schilling R.L., Song R., Vondracek Z.: Bernstein Functions. Theory and Applications. De Gruyter, Berlin (2012)
Slade G., Tomberg A.: Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component \({|\varphi|^4}\) model. Commun. Math. Phys. 342, 675–737 (2016)
Sokal A.D.: A rigorous inequality for the specific heat of an Ising or \({\varphi^4}\) ferromagnet. Phys. Lett. 71, 451–453 (1979)
Suzuki M., Yamazaki Y., Igarashi G.: Wilson-type expansions of critical exponents for long-range interactions. Phys. Lett. 42, 313–314 (1972)
Wilson K.G., Fisher M.E.: Critical exponents in 3.99 dimensions. Phys. Rev. Lett. 28, 240–243 (1972)
Yosida K.: Functional Analysis, 6th edn. Springer, Berlin (1980)
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Slade, G. Critical Exponents for Long-Range \({O(n)}\) Models Below the Upper Critical Dimension. Commun. Math. Phys. 358, 343–436 (2018). https://doi.org/10.1007/s00220-017-3024-5
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DOI: https://doi.org/10.1007/s00220-017-3024-5