Question 25: Let n and N be positive integers, and let \mathcal{F} be…

Let $n$ and $N$ be positive integers, and let $\mathcal{F}$ be an arbitrary nonempty family of subsets of the universe $\{1, \ldots, n\}$. Suppose each element $x \in \{1, \ldots, n\}$ in the universe receives an integer weight $w(x)$, each of which is chosen independently and uniformly at random from $\{1, \ldots, N\}$. The weight of a set $S$ in $\mathcal{F}$ is defined as

$w(S) = \sum_{x \in S} w(x)$

We want to explore the probability of the following "good" event: there is a unique set in $\mathcal{F}$ that has the minimum weight among all sets of $\mathcal{F}$.

Based on the above data, answer the given subquestions.

Consider the problem of finding perfect matchings in a simple, undirected graph. Recall that the Tutte matrix of a graph $G$, denoted $Z_G$, is given by:

$Z_G[i, j] = \begin{cases} z_{ij} & \text{if } (i, j) \text{ is an edge and } i < j \\ -z_{ji} & \text{if } (i, j) \text{ is an edge and } j > i \\ 0 & \text{if } (i, j) \text{ is not an edge} \end{cases}$

Suppose each edge is assigned a random weight in $\{1, \ldots, 2m\}$, and $\mathcal{F}$ is the set of perfect matchings. Further, let us replace each indeterminate $z_{ij}$ in the Tutte matrix of the graph is replaced with $2^{w_{ij}}$ where $w_{ij}$ is the randomly assinged weights of the edge $(i, j)$ from above.

If $G$ has no perfect matching, what is the determinant of $Z$ with the variables substituted for these weights?