Abstract
We study the four-dimensional n-component \({|\varphi|^4}\) spin model for all integers \({n \ge 1}\) and the four-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case \({n=0}\) interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order p, and prove the existence of a logarithmic correction to mean-field scaling, with power \({\frac 12\frac{n+2}{n+8}}\), for all \({n \ge 0}\) and \({p > 0}\). The proof is based on an improvement of a rigorous renormalisation group method developed previously.
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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.: 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)
Brézin E., Le Guillou J.C., Zinn-Justin J.: Approach to scaling in renormalized perturbation theory. Phys. Rev. D. 8, 2418–2430 (1973)
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., 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)
Fernández R., Fröhlich J., Sokal A.D.: Random walks, critical phenomena, and triviality in quantum field theory. Springer, Berlin (1992)
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., 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)
Larkin, A.I., Khmel’Nitskiĭ, D.E.: Phase transition in uniaxial ferroelectrics. Soviet Physics JETP 29, 1123–1128 (1969) (English translation of Zh. Eksp. Teor. Fiz. 56, 2087–2098 (1969))
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)
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)
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Communicated by Abdelmalek Abdesselam.
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Bauerschmidt, R., Slade, G., Tomberg, A. et al. Finite-Order Correlation Length for Four-Dimensional Weakly Self-Avoiding Walk and \({|\varphi|^4}\) Spins. Ann. Henri Poincaré 18, 375–402 (2017). https://doi.org/10.1007/s00023-016-0499-0
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DOI: https://doi.org/10.1007/s00023-016-0499-0