The density of integral quadratic forms having a $k$-dimensional totally isotropic subspace

We investigate the probability that a random quadratic form in $\mathbb{Z}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ has a totally isotropic subspace of a given dimension. We show that this global probability is a product of local probabilities. Our main result computes these local probabilities for quadratic forms over the $p$-adics. The formulae we obtain are rational functions in $p$ invariant upon substituting $p\mapsto 1\mathrm{/}p$.