Precision check on the triviality of the ${\varphi }^4$ theory by a new simulation method

We report precise simulations of ${\varphi }^4$ theory in the Ising limit. A recent technique to stochastically evaluate the all-order strong coupling expansion is combined with exact identities in the closely related Aizenman random current representation. In this way estimates of the renormalized coupling close to the continuum limit become possible with unprecedented precision and yet low CPU cost. As a sample application we present results for the unbroken phase of the Ising model in dimensions 3, 4 and 5 and investigate the question of triviality by studying a finite size scaling continuum limit.

Figure 1

All measured couplings with fits (full and dashed lines) discussed in the text. Error bars are barely visible within the symbols.

Figure 2

Numerical couplings together with the renormalization group evolution starting from the finest lattice. Dashed lines include lattice artifacts of the $\beta$ function.

Figure 3

The order in $\beta$ of the diagrams sampled for $L=32$, $D=4$ ($u=v$ only).