  We prove two results that shed new light on the monotone complexity of the
spanning tree polynomial, a classic polynomial in algebraic complexity and
beyond.
  First, we show that the spanning tree polynomials having $n$ variables and
defined over constant-degree expander graphs, have monotone arithmetic
complexity $2^{\Omega(n)}$. This yields the first strongly exponential lower
bound on the monotone arithmetic circuit complexity for a polynomial in VP.
Before this result, strongly exponential size monotone lower bounds were known
only for explicit polynomials in VNP (Gashkov-Sergeev'12, Raz-Yehudayoff'11,
Srinivasan'20, Cavalar-Kumar-Rossman'20, Hrubes-Yehudayoff'21).
  Recently, Hrubes'20 initiated a program to prove lower bounds against general
arithmetic circuits by proving $\epsilon$-sensitive lower bounds for monotone
arithmetic circuits for a specific range of values for $\epsilon \in (0,1)$. We
consider the spanning tree polynomial $ST_{n}$ defined over the complete graph
on $n$ vertices and show that the polynomials $F_{n-1,n} - \epsilon \cdot
ST_{n}$ and $F_{n-1,n} + \epsilon \cdot ST_{n}$ defined over $n^2$ variables,
have monotone circuit complexity $2^{\Omega(n)}$ if $\epsilon \geq
2^{-\Omega(n)}$ and $F_{n-1,n} = \prod_{i=2}^n (x_{i,1} +\cdots + x_{i,n})$ is
the complete set-multilinear polynomial. This provides the first
$\epsilon$-sensitive exponential lower bound for a family of polynomials inside
VP. En-route, we consider a problem in 2-party, best partition communication
complexity of deciding whether two sets of oriented edges distributed among
Alice and Bob form a spanning tree or not. We prove that there exists a fixed
distribution, under which the problem has low discrepancy with respect to every
nearly-balanced partition. This result could be of interest beyond algebraic
complexity.
