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.



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


