larger and is about $N^g_{auto} \sim 1500$ Metropolis steps. (The correlation between the values of the gauge field is less than $3 \%$ for the configurations separated by $N^g_{auto}$ Metropolis steps. Each metropolis step consists of the renewing the fields over all the lattice.) The autocorrelation time for the scalar field is essentially smaller than for the gauge fields and is of the order of $N^{\phi}_{auto} \sim 20$. The estimated time for preparing the equilibrium starting from the cold start far from the phase transition within the Higgs phase is about $18000$ Metropolis steps for the considered values of couplings. At the same time near the phase transition and within the symmetric phase the estimated time for preparing the equilibrium is up to $3$ times larger.

The tree level estimates of lattice quantities
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At finite $\lambda$ the line of constant renormalized $\alpha$ is not a line of constant physics, because the mass of the Higgs boson depends on the position on this line. Thus, in order to investigate the line of constant physics one should vary $\lambda$ together with $\gamma$ to keep the ratio of lattice masses $M_H/M_W$ constant.

In order to obtain the tree level estimates let us 