The semantics of deletion
“Remove component $i$” means different things in different fields — and the two cavity identities answer different removal questions.
The algebraic similarity between the multiplicative cavity (divide out a survival function) and the rank-one Schur cavity (subtract a mediated interaction) is what started this project. But similarity of formulas can obscure the physical question: the interventions they price are not the same. This page fixes the vocabulary used across the site.
Marginal vs conditional
Let the latent performances be jointly Gaussian, $X \sim N(\mu, \Sigma)$.
| Intervention | Surviving distribution | Which cavity |
|---|---|---|
| Scratch competitor $i$ | the marginal: $X_{-i} \sim N(\mu_{-i},\, \Sigma_{-i,-i})$ — the covariance is simply the submatrix | multiplicative (races) |
| Pin / condition $X_i = c$ | the conditional: $\Sigma_{-i,-i} - \Sigma_{-i,i}\,\Sigma_{ii}^{-1}\,\Sigma_{i,-i}$ | Schur / rank-one |
| Delete a coordinate of the precision $A = \Sigma^{-1}$ | $\left(A_{-i,-i}\right)^{-1}$ — equal to the conditional covariance above | Schur / rank-one |
Scratching uses a marginal; Schur uses a conditional. A correlated race does not automatically invoke a Schur complement when a competitor is scratched — the field simply loses a coordinate, correlations among survivors unchanged. The Schur complement enters when the intervention pins, conditions, or re-equilibrates the field. The fast correlated transform combines the identities in the legitimate way: conditioning on latent factors (Schur side), then scratching within the conditionally independent race (multiplicative side).
The zoo of removals
- Scratch a candidate — a competitor leaves the race; survivors' joint law is the marginal. (Experiments 1, 3, 6.)
- Suppress a hazard — a transition channel's intensity is removed from the total; the background dynamics continue unchanged. (The blocked-window counterfactual, read kinetically.)
- Pin a coordinate — a degree of freedom is clamped; the rest re-equilibrate around the constraint. This is what experiment 2 computes (its source says “pinning” deliberately).
- Remove bonds — a physical spring/edge disappears: $H' = H - k\,vv^\top$, a Woodbury update, not a principal submatrix.
- Remove a material site (vacancy) — all incident bonds vanish (small-rank block update) and the neighborhood relaxes, generally anharmonically.
- Recompute the background — the intervention changes the environment that generates the rates (temperature, stress); nothing above applies directly. (Experiments 4–5 showed this axis is orthogonal.)
Each row of this list is a different physical question with a different cheap identity (submatrix, Schur complement, Woodbury, block downdate) or none at all. Claims on this site name the intervention explicitly.