Convergence of probability measures which (asymptotically) concentrate along a submanifold

Question

Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define

\begin{align} P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\ z (\beta) &= \int \exp \left( - \beta V ( y ) \right) \, dy \end{align}

with the assumption that $V$ is sufficiently well-behaved at the boundary that $z (\beta) < \infty$ for all $\beta > 0$.

Now, if $V$ has a unique global minimiser at (wlog) $V \left(\mathbf{0}\right) = 0$, one can usually argue that as $\beta \to \infty$, $P_\beta$ converges in law to a delta measure at $\mathbf{0}$, maybe with some extra conditions in place.

My situation is that $V$ takes its minimum value (again, wlog taken to be $0$) along a codimension-1 submanifold, i.e. along

$$\mathcal{F} = \{ x \in (-1, 1)^d : V(x) = 0 \}.$$

Now, I would like to reason that, as $\beta \to \infty$, $P_\beta$ converges in law to some measure which concentrates along $\mathcal{F}$. I would guess that the answer has something to do with the Hausdorff measure on $\mathcal{F}$, but i) my intuition for such matters is not very strong, and ii) even if it were, I am not sure where I would look for the relevant mathematical tools to prove it.

As such, my questions are:

1. What is a reasonable conjecture for the limiting behaviour of $P_\beta$ as $\beta \to \infty$, and
2. How can I prove it?

For 1., I'd like any conjecture to come equipped with some justification, or a relevant example to which I can compare things; for 2., if a full proof would take too long to outline, a relevant reference would be appreciated.

Answer 1

Here is an outline of the proof. Let $n:=d$. Suppose that for some real $\delta>0$ the $\delta$-neighborhood of the set $F:=\mathcal F$ can be covered by pairwise disjoint sets $U_1,\dots,U_k$ such for each $j=1,\dots,k$ the boundary of the set $U_j$ is of zero Lebesgue measure and the closure $\bar U_j$ of $U_j$ can be parameterized by a smooth enough map $$[-1,1]^{n-1}\times[-1,1]\ni(s,t)\mapsto x_j(s,t)\in\bar U_j$$ so that the map $$[-1,1]^{n-1}\ni s\mapsto x_j(s,0)\in F\cap\bar U_j$$
is onto, that is, a smooth enough parameterization of the "$j$th piece" $F\cap\bar U_j$ of the set $F$. Let $$J_j(s,t):=\partial x(s,t)/\partial(s,t)$$ be the corresponding Jacobian determinant.

Next, let $$W_j(s,t):=V(x_j(s,t)).$$ Then $W_j(s,t)\ge W_j(s,0)$ and $W_j(s,t)\sim W''_{j;tt}(s,0)t^2/2$ as $t\to0$, assuming $W''_{j;tt}(s,0)\ne0$ and hence $W''_{j;tt}(s,0)>0$ for all $s$, where $W''_{j;tt}$ is the second partial derivative of $W_j$ in $t$. So, by standard reasoning, for any smooth enough real-valued function $f$ on $(-1,1)^n$ and $$g_j(s,t):=g_{f;j}(s,t):=f(x_j(s,t)),$$ we have $$\int_{U_j}dx\,f(x)\exp\{-b^2 V(x)\} \\ =\int_{[-1,1]^{n-1}}ds\,\int_{[-1,1]}dt\,|J_j(s,t)|g_j(s,t) \exp\Big\{-\frac{b^2 W''_{j;tt}(s,0)t^2}{2+o(1)}\Big\} \\ \sim\frac{\sqrt{2\pi}}b\, \int_{[-1,1]^{n-1}}ds\,\frac{|J_j(s,0)|g_j(s,0)}{\sqrt{W''_{j;tt}(s,0)}} $$ as $b\to\infty$. It follows that the probability measure $P_{b^2}$ converges to the probability measure $P_\infty$ given by the condition $$\int f\,dP_\infty=\sum_{j=1}^k\int_{[-1,1]^{n-1}}\frac{ds\,|J_j(s,0)|f(x_j(s,0))}{\sqrt{W''_{j;tt}(s,0)}}\Big/\sum_{j=1}^k\int_{[-1,1]^{n-1}}\frac{ds\,|J_j(s,0)|}{\sqrt{W''_{j;tt}(s,0)}}$$ for all nice enough $f$.