An introduction

Races, cavities, and the observation that one global solve contains every leave-one-out answer.

Kinetics, at the resolution a simulator sees it, is a race. A system sits in a state with $N$ available escape channels — reactions, hops, bond ruptures, local rearrangements — and whichever channel fires first determines the next state. Kinetic Monte Carlo runs this race with exponential clocks; molecular dynamics runs it with the true (expensive) physics. Either way, the data the simulation produces is race data: which channel won, and occasionally by how much. This page sets out the model we use for such races, the two leave-one-out identities that make it fast, and why those identities are the cavity method of statistical mechanics wearing algorithmic clothes.

The race model

Give each competitor $i$ a latent propensity $\mu_i$ and a performance $X_i = \mu_i + \epsilon_i$, with the $\epsilon_i$ independent draws from a base density. The lowest performance wins (read $X_i$ as a first-passage time, an activation barrier crossed, or simply “performance” with sign flipped). The probability that $i$ wins is

$p_i = \displaystyle\int f_i(x) \prod_{j \neq i} S_j(x)\, dx$

where $f_i$ is the density of $X_i$ and $S_j = 1 - F_j$ the survival function of $X_j$. This is Thurstone's 1927 model of comparative judgment, and it is also exactly the structure of a first-event problem in kinetics. Two special cases anchor the landscape:

Proportional hazards ⇒ Luce. If $X_i \sim \text{Exp}(k_i)$ then $p_i = k_i / \sum_j k_j$ — the Luce/Harville formula, and precisely the transition rule of kinetic Monte Carlo. But exponentiality is sufficient, not necessary: with cumulative hazards $H_i(t) = a_i H_0(t)$ for any common baseline $H_0$, the win probability is exactly $a_i / \sum_j a_j$. An arbitrary common time change of exponential clocks still gives Luce, and deleting a competitor still rescales the others proportionally (IIA).
What actually breaks IIA. Not non-exponentiality per se, but differential structure: channels with different hazard shapes, history dependence, statistical dependence, or different loadings on a shared environment. A common perturbation of all hazards cancels exactly at first order. The Thurstone race with a general base density is one tractable model of differential shape; correlated races model shared-environment loadings (deletion semantics explains which removal question each answers).

The inverse problem is the one that matters for surrogates: given observed winning frequencies $p_i$ (from expensive simulation), recover the latent propensities $\mu_i$. With $\mu$ in hand, the model answers questions the raw frequencies cannot: what happens to the race if a channel is blocked, if the field shrinks or grows, if two systems' channels are pooled. That inversion is the fast ability transform, implemented in thurstone.

Identity one: the multiplicative cavity

Computing all $N$ win probabilities looks like $N$ integrals, each against a product of $N-1$ survival functions: $O(N^2)$ work on a lattice. The trick is to build the field once. The survival function of the minimum of the whole field is a single product,

$S_{\text{field}}(x) = \prod_{j} S_j(x)$

and the field faced by competitor $i$ — everyone except $i$ — is recovered by dividing $i$ back out:

$S_{-i}(x) = \dfrac{S_{\text{field}}(x)}{S_i(x)}$

One pass builds $S_{\text{field}}$; then every one of the $N$ leave-one-out fields costs a pointwise division. Total cost $O(N)$ per lattice sweep, and this is what makes the ability transform practical for fields of thousands (Cotton 2021; see rest_min_cdf in the thurstone source).

Identity two: the rank-one cavity

Now the quadratic analogue. Suppose the system is Gaussian — a harmonic approximation around equilibrium, an elastic network, a Gaussian field — with energy $H(x) = \tfrac12 x^\top A x$ and inverse $G = A^{-1}$ (the covariance, susceptibility, or Green function). Deleting site $i$ means inverting the principal submatrix $A_{-i,-i}$. Naively that is $n$ separate $O(n^3)$ inversions. But the full inverse already contains every answer:

$\left(A^{-1}_{-i,-i}\right)_{jk} \;=\; G^{(i)}_{jk} \;=\; G_{jk} - \dfrac{G_{ji}\,G_{ik}}{G_{ii}}, \qquad j,k \neq i$

One factorization, then each deletion is a rank-one downdate: $O(n^2)$ for the whole cavity matrix, and $O(1)$ for any single entry. The subtracted term $G_{ji}G_{ik}/G_{ii}$ has a clean physical reading — it is the part of the interaction between $j$ and $k$ that is mediated through $i$. Removing $i$ removes exactly the mediated component.

Statistical mechanics knows this identity as the cavity Green function: it is the engine of the cavity method for spin glasses and of the Schur-complement arguments behind local laws in random matrix theory. Statistics knows its scalar shadow as the regression leave-one-out identity $\hat y^{(-i)}_i = y_i - e_i/(1 - h_{ii})$. What appears underexploited is the computational framing: statistical mechanics typically reasons “choose a site, construct its cavity system, derive a local quantity,” whereas the algorithmic reading is

invert once — and obtain every single-site cavity system essentially for free.

For scalar observables the economics get better still: many leave-one-out quantities collapse to formulas in $G_{ii}$ and a few inner products, so you often do not even pay $O(n^2)$ per deletion. And the identity iterates: for a subset $S$ of deletions,

$G^{(S)} = G_{\bar S, \bar S} - G_{\bar S, S}\, G_{S,S}^{-1}\, G_{S, \bar S}$

so leave-$k$-out costs only a $k \times k$ inversion. For $k = 1, 2, 3$ — single defects, defect pairs, small clusters — that is extremely cheap. One global inverse is a compressed representation of an enormous family of defect and cavity systems.

The dictionary

Racing	Kinetics / statistical mechanics	Shared object
Competitor $i$	Escape channel, reaction pathway, weak bond, soft mode	A latent process that may fire first
Ability $\mu_i$	Log-rate, negative activation barrier, defect propensity	Latent scale with meaningful gaps
Race outcome	First event: which channel fired	Extremal (argmin) observation
Betting odds	Empirical channel frequencies from simulation	Win probabilities $p_i$
Scratching a horse	Blocking a channel, pinning a site, removing a defect	Leave-one-out downdate
Ability transform	Kinetic surrogate: frequencies $\to$ propensities $\to$ predictions	Inverse problem over the race model

Why this might matter

Kinetic Monte Carlo is already a race — but an exponential one, with rates supplied a priori. The proposal here runs the other way and drops the exponential crutch: observe races run by the true expensive physics, invert them through a Thurstone-style model with a realistic noise density, and obtain latent propensities that (a) transfer to fields the simulation never ran — channels added, blocked, or re-pooled — and (b) come with the entire leave-one-out ensemble attached, at the price of one global computation. Where the quadratic structure is available too, the rank-one cavity answers “what if each site were absent?” for all sites at once — every single-defect susceptibility from one solve.

The research program turns this into concrete questions; the applications page ranks candidate domains; and the experiments verify both identities numerically, with timings.