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.