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I'm interested in an integral of the form

$$\int d\vec{\theta} e^{-\beta (\vec{\theta}-\vec{\theta}_0) \cdot K \cdot (\vec{\theta}-\vec{\theta}_0)} \delta(f(\vec{\theta}))$$

where $\vec{\theta}$ represents a set of real numbers, and the integration is implied to be over their entire domain, i.e. $\mathbb{R}^N$, not an indefinite integral. $K$ is a constant square matrix and $\beta$ a constant scalar. In principle $f$ is an arbitrary but well-behaved function. Although a general solution to such an integral would be great, I'm guessing that is not feasible. Perhaps an answer to the following subquestions is:

1) Suppose $f$ is a linear function, i.e. the integral becomes

$$\int d\vec{\theta} e^{-\beta (\vec{\theta}-\vec{\theta}_0) \cdot K \cdot (\vec{\theta}-\vec{\theta}_0)} \delta(M\cdot\vec{\theta} - \vec{c}).$$

My intuition tells me that such a linear constraint should lead to only linear terms in the exponential, meaning we should somehow be able to rewrite the integral as

$$\int d\vec{\phi} e^{-\beta (\vec{\phi}-\vec{\phi}_0) \cdot K' \cdot (\vec{\phi}-\vec{\phi}_0) + \beta J \cdot \vec{\phi}}$$

where $\vec{\phi}$ is a vector containing a subset of the $\theta$ degrees of freedom, the rest having been integrated out using the delta function, and $K'$ is similarly a reduced version of $K$.

Am I correct in this conjecture, and is it possible to relate $K'$ and $J$ to $K$, $M$ and $\vec{c}$ in closed form?

2) I am primarily interested in how the integral scales with $\beta$. Without constraints, the scaling is something like $\beta^{-N/2}$ with $N$ the number of degrees of freedom. Introducing a linear constraint, if I'm correct in my first point, changes this scaling to $e^{\beta}\beta^{-N'/2}$ with $N'$ the number of remaining degrees of freedom after integrating over the constraints.

Can anything be surmised as to this scaling for arbitrary constraints, or at least for polynomial constraints of higher order?

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