Controlled shrinking energy-density shell and three-ensemble equivalence

Let $C_L$ be the periodic exact-$N_L$ canonical law. For $\mathrm{\mid }t\mathrm{\mid }<\beta$, $E_{C_L}{e}^{t{H}_{\mathrm{p}\mathrm{e}\mathrm{r}}}=Z_{\mathrm{p}\mathrm{e}\mathrm{r}}(L,N_L;\beta -t)\mathrm{/}{Z}_{\mathrm{p}\mathrm{e}\mathrm{r}}(L,N_L;\beta )\mathrm{.}$ By [F1], the normalized log moment generating function converges to $F_{\beta -t}(\rho )-F_{\beta }(\rho )$, whose derivative at zero is $e$. For any fixed $ϵ>0$, choose small positive and negative $t$ so the Chernoff exponents for $H\mathrm{/}{L}^d\ge e+ϵ$ and $H\mathrm{/}{L}^d\le e-ϵ$ are strictly negative. Hence the canonical probability outside the fixed $ϵ$-window tends to zero.

The supporting activity and singleton minimizer are given. The exact-$N$ periodic rate is [F1]; the choices in [A1], the superstable regular potential, and the empirical-field convention [F4] are precisely the other hypotheses of [F5]. Applying [F5] with periodic boundary gives, for every open $U\ni P_{\ast }$, a constant $c_U>0$ with $\mathrm{lim\hspace{0.17em}sup}{L}^{-d}\log C_L^{\ast }(R_{L,\omega }\mathrm{\notin }U)\le -c_U$. This supplies the exponential estimate needed for the shell comparison.

Choose strictly increasing thresholds $L_k\ge k$ so that for every $L\ge L_k$ the probability outside the $1\mathrm{/}k$ window is at most $1\mathrm{/}k$. Set $k(L)=\max \{k:L_k\le L\}$ and ${\delta }_L=1\mathrm{/}k(L)$. Then ${\delta }_L\downarrow 0$ and the canonical probability of the selected shell tends to one. Positivity of the canonical weight on the shell implies positive Poisson reference measure, so the shell law is defined eventually.

Let $U$ be any open ${\tau }_L$ neighborhood of $P_{\ast }$ and let $S_L=\{N=N_L,\mathrm{\mid }{H}_{L,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{L}^d-e\mathrm{\mid }\le {\delta }_L\}$ be the selected shell event. On $S_L$, the unnormalized periodic Boltzmann factor lies between $e^{-\beta L^d(e+{\delta }_L)}$ and $e^{-\beta L^d(e-{\delta }_L)}$. For each measurable event, the normalization constants cancel when comparing conditional probabilities; taking infima over measurable supersets gives the same inequality for outer probabilities, so $(Q_{{\mathrm{\Lambda }}_L}(\cdot \mid S_L){)}^{\ast }(R_{L,\omega }\mathrm{\notin }U)\le \frac{e^{2\beta {\delta }_L{L}^d}}{C_L^{\mathrm{p}\mathrm{e}\mathrm{r}}(S_L)}(C_L^{\mathrm{p}\mathrm{e}\mathrm{r}}{)}^{\ast }(R_{L,\omega }\mathrm{\notin }U)\mathrm{.}$ By step 2.1, $C_L^{\mathrm{p}\mathrm{e}\mathrm{r}}(S_L)\to 1$. By step 1.2, the last outer probability has a strictly negative exponential rate. Since ${\delta }_L\to 0$, the right side tends to zero. Thus the uniform shell empirical field converges in probability to $P_{\ast }$.

For any open neighborhood $U$ of $P_{\ast }$, choose a basic neighborhood $V$ of $P_{\ast }$ contained in $U$, defined by finitely many strict tame-expectation inequalities. It is evaluation-measurable. The free exact-$N$ empirical fields concentrate on $V$ by [F2], so their outer probabilities outside $U$ vanish. The periodic grand empirical fields obey the good LDP [L1]; its unique zero is $P_{\ast }$ by [F3], so the measurable-set upper bound outside $V$ has a strictly negative rate by compactness of rate sublevels. As $U^c\subseteq V^c$, its outer probability tends to zero. Together with steps 2.1 and 3.1, this proves the stated three-ensemble conclusion.