TY - JOUR

T1 - Logarithmic Correction for the Susceptibility of the 4-Dimensional Weakly Self-Avoiding Walk

T2 - A Renormalisation Group Analysis

AU - Bauerschmidt, Roland

AU - Brydges, David C.

AU - Slade, Gordon

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2015/7/1

Y1 - 2015/7/1

N2 - We prove that the susceptibility of the continuous-time weakly self-avoiding walk on (Formula presented.), in the critical dimension d = 4, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent (Formula presented.) for the logarithm. The susceptibility has been well understood previously for dimensions d ≥ 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.

AB - We prove that the susceptibility of the continuous-time weakly self-avoiding walk on (Formula presented.), in the critical dimension d = 4, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent (Formula presented.) for the logarithm. The susceptibility has been well understood previously for dimensions d ≥ 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.

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U2 - 10.1007/s00220-015-2352-6

DO - 10.1007/s00220-015-2352-6

M3 - Article

AN - SCOPUS:84937761342

SN - 0010-3616

VL - 337

SP - 817

EP - 877

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 2

ER -