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Abstract |
We will discuss the asymptotic behaviour of the critical two-point function for a long-range version of the n-component $|varphi|^4$ model and the weakly self-avoiding walk (WSAW) on the d-dimensional Euclidean lattice with d=1,2,3. The WSAW corresponds to the case n=0 via a supersymmetric integral representation. We choose the range of the interaction so that the upper-critical dimension of both models is $d+epsilon$. Our main result is that, for small $epsilon$ and small coupling strength, the critical two-point function exhibits mean-field decay, confirming a prediction of Fisher, Ma, and Nickel. The proof makes use of a renormalisation group method of Bauerschmidt, Brydges, and Slade, as well as a cluster expansion. This is joint work with Martin Lohmann and Gordon Slade. |
