Scaling limits and critical behaviour of the 4-dimensional n-component $|\varphi|^4$ spin model
| Autor: | Bauerschmidt, Roland, Brydges, David C., Slade, Gordon |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | J. Stat. Phys, 157:692--742, (2014) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10955-014-1060-5 |
| Popis: | We consider the $n$-component $|\varphi|^4$ spin model on $\mathbb{Z}^4$, for all $n \geq 1$, with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent $\frac{n+2}{n+8}$ for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for $n =1,2,3$; double logarithmic scaling for $n=4$; and is bounded when $n>4$. In addition, for the model defined on the $4$-dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the $|\varphi|^4$ model. Comment: 55 pages |
| Databáze: | arXiv |
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