This could be \(v_{boom}\) a phenomenon where a voltage \(V\) between a gap of length \(d\) span an electric force \(E=\cfrac{V}{d}\) that accelerates free ions to \(v_{boom}\) and generates basic particles.
Intuitively, a higher voltage is needed for small gap to accelerate the particles to \(v_{boom}\) before covering the length \(d\),
\(v^2=2ax\)
\(x\le d\)
from \(v^2=u^2+2ax\),
where \(a\) is the acceleration under the constant field \(E=\cfrac{V}{d}\) and we assume that the particle has zero initial velocity.
\(F=qE\)
so,
\(a=\cfrac{F}{m_i}=\cfrac{qE}{m_i}=\cfrac{qV}{m_id}\)
thus,
\(v^2=2ax=2\cfrac{qV}{m_i}\cfrac{x}{d}\)
If \(v=v_{boom}=3.4354*\cfrac{density}{Z}\) and
\(PV_{ol}=nRT\)
\(P=\cfrac{n}{V_{ol}}RT=\cfrac{M}{m_iN_AV_{ol}}RT=\cfrac{density}{m_i}kT\)
where \(N_A\) is the number per mole, \(M\) is total mass of the gas. So
\(v=v_{boom}=3.4354*\cfrac{m_iP}{ZkT}\)
and
\(\left(3.4354*\cfrac{m_i}{Zk}\right)^2\left(\cfrac{P}{T}\right)^2=2\cfrac{qV}{m_i}\cfrac{x}{d}\)
\(V\cfrac{x}{d}=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}\left(\cfrac{P}{T}\right)^2\)
\(x\le d\)
\(V=V_{min}\) when
\(x=d\)
\(V_{min}=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}\left(\cfrac{P}{T}\right)^2\)
\(V_{min}=A\left(\cfrac{P}{T}\right)^2\)
where \(A=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}\)
This assumes that after attaining \(v_{boom}\), collisions that generates particles is a certainty before reaching the other electrode, which may not be true for very low pressure and small gap length.
Strictly speaking, a discharge occurs,
\(V\gt V_{min}\)
\(V\gt A\left(\cfrac{P}{T}\right)^2\)
where \(A=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}\)
Maybe...
