Total-energy concentration and controlled canonical windows

Integrating the marks in Interacting marked phase-space and reference measures gives the canonical marked partition as ${\kappa }_{b^{\mathrm{\prime }}}^{N_n}{Z}_{L_n,N_n,\mathrm{p}\mathrm{e}\mathrm{r}}^{\mathrm{p}\mathrm{o}\mathrm{s}}(b^{\mathrm{\prime }})$. The exact-$N_n$ positional sector of the unit-intensity Poisson reference contributes the common factor $e^{-V_n}$ relative to the Lebesgue $1\mathrm{/}{N}_n!$ partition; it cancels in the ratio of marked partition functions. By [F1], for each fixed $t$ with $b-t>0$, ${\lim }_{n\to \mathrm{\infty }}{V}_n^{-1}\log E_{C_n^{\mathrm{p}\mathrm{h}}}{e}^{t\mathcal{H}}=\mathrm{\Lambda }(t):=F_{b-t}(\rho )-F_b(\rho )+\rho \log ({\kappa }_{b-t}\mathrm{/}{\kappa }_b),$ by [F1, F2]. Differentiability and the explicit Gaussian factor give $\mathrm{\Lambda }(0)=0$ and ${\mathrm{\Lambda }}^{\mathrm{\prime }}(0)=-F_b^{\mathrm{\prime }}(\rho )+d\rho \mathrm{/}(2b)=e_b(\rho )$.

Fix $\epsilon >0$. By the derivative at zero, choose a sufficiently small fixed $t>0$ so $\mathrm{\Lambda }(t)-t(e_b+\epsilon )<0$. On the upper event, $1\le e^{t(\mathcal{H}-V_n(e_b+\epsilon ))}$, so integrating this pointwise inequality gives $C_n^{\mathrm{p}\mathrm{h}}(\mathcal{H}\mathrm{/}{V}_n\ge e_b+\epsilon )\le e^{-t{V}_n(e_b+\epsilon )}{E}_{C_n^{\mathrm{p}\mathrm{h}}}{e}^{t\mathcal{H}};$ its logarithm divided by $V_n$ tends to that strictly negative number, so it is at most $-c_{+}{V}_n$ in the exponent for some $c_{+}>0$ and all large $n$. Choose a sufficiently small fixed $t<0$ instead; on $\{\mathcal{H}\mathrm{/}{V}_n\le e_b-\epsilon \}$ one has $e^{t\mathcal{H}}\ge e^{t{V}_n(e_b-\epsilon )}$, so integrating this lower-tail pointwise bound gives an exponent with limit $\mathrm{\Lambda }(t)-t(e_b-\epsilon )<0$. Taking the smaller positive decay constant proves the two-sided bound.

For each integer $j\ge 1$, the fixed-width conclusion of step 2.1 supplies an integer $n_j$ increasing strictly such that for all $n\ge n_j$ the probability outside width $1\mathrm{/}j$ is at most $1\mathrm{/}j$. Let $j(n)=\max \{j:n_j\le n\}$ for $n\ge n_1$, and put ${\delta }_n=1\mathrm{/}j(n)$; assign any positive nonincreasing values before $n_1$. Then $j(n)\to \mathrm{\infty }$, so ${\delta }_n\downarrow 0$, and the probability outside $A_n$ tends to zero. Since the canonical density is strictly positive at every finite-energy point, $C_n^{\mathrm{p}\mathrm{h}}(A_n)>0$ for large $n$ implies positive reference measure. On $A_n$, $0\le \sum K(p_x)\le V_n(e_b+{\delta }_n)+B{N}_n$ by stability [F3]; the momenta thus lie in a bounded Euclidean ball, while positions lie in a finite cube. The sigma-finite product reference in Interacting marked phase-space and reference measures assigns finite measure to that bounded sector, so the band has finite reference measure.