TY - JOUR

T1 - Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk

AU - Bauerschmidt, Roland

AU - Brydges, David C.

AU - Slade, Gordon

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

PY - 2015/8/1

Y1 - 2015/8/1

N2 - We prove (Formula Presented.) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on (Formula Presented.), in the upper critical dimension d = 4. This is a statement that the critical exponent η exists and is equal to zero. Results of this nature have been proved previously for dimensions d≥5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.

AB - We prove (Formula Presented.) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on (Formula Presented.), in the upper critical dimension d = 4. This is a statement that the critical exponent η exists and is equal to zero. Results of this nature have been proved previously for dimensions d≥5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.

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U2 - 10.1007/s00220-015-2353-5

DO - 10.1007/s00220-015-2353-5

M3 - Article

AN - SCOPUS:84929085424

SN - 0010-3616

VL - 338

SP - 169

EP - 193

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 1

ER -