An Interactive Lab for Peeling Thresholds and Failure-Conditioned Measurements
Degree-one peeling runs live in your browser on a random k-uniform hypergraph with M vertices (cells) and d edges (keys). Move the controls to re-simulate. Red runs got stuck (peeling failed); blue runs peeled every edge.
Fewest degree-one cells near t*–median over successful runs
Critical window M−1/4–
ADegree-one cells over the whole run
succeededfailedfluid limit ℓ(t)
Height is the fraction of cells holding exactly one key, the fuel for the next peeling step. For large M every path follows the deterministic fluid curve (law of large numbers). At r = 0 the curve touches zero at t*; that tangency is what defines the threshold αk.
BZoom into the bottleneck
succeededfailedM·ℓ / √M
Time is stretched by M1/4 and height by √M. In this scaling the local shape converges to a parabola as M grows. Near t* the paths look like the parabola shifted up or down as a whole; a downward shift reaches zero and the run fails. That vertical shift is the score R of the proof, and failure ≈ {R ≤ 0}.
CDrift-corrected fluctuations of V
succeededfailedmeanmean drift correction
Vs = Σj fj(ac/a(s)) Uj(s) weights the degree profile so that its leading drift cancels. Plotted is the approximately compensated fluctuation Vs − V0 − AVs; the dashed line is the first-order drift correction AV, which stays near zero on this scale. High V means few degree-one cells at t*, so red paths end high.
DTwo sources of randomness
sd of corrected fluctuationsd of score Rτkσk
R = (initial randomness of the graph) + (noise collected while peeling). The score already varies at t = 0; the corrected fluctuation starts at zero and accumulates. In the limit σk² = (initial variance) + τk², with both constants evaluated at t*. Curves stop once more than 1% of runs have failed, since the survivors are a selected sample.
Notes
Notation follows the accompanying working paper: Qn is the number of degree-one vertices after n peeled edges, t = n/M, xc, ac, βk, τk, σk are the closed-form constants. Every black curve comes from the analytic formulas; nothing is fitted to the simulation.
The drift correction uses a first-order uniform-partner transition. It is an approximation to the Doob compensator, not the exact finite-graph conditional drift, especially for Plain mappings. The 99% survival cutoff limits but does not remove survivor selection.
Finite-size failure rates can differ from the limiting value. A threshold-center correction of order M−1/6 in the window coordinate is known for the configuration setting; this page does not assert a finite-size error bound for Plain mappings.
Out of scope: k = 2, rates of convergence, the rare-failure regime far below the threshold.
Part II
Failure-conditioned measurement and sign imbalance
Before peeling, the sketch can be summarized by its centered count energy, standardized to X. For every fixed sign vector X has mean 0 and variance 1 over Plain mappings. Peeling failure depends only on the unsigned mapping, so the signs cannot change which runs fail. They do change how strongly X is correlated with the failure score R. In the limit this correlation is ηkρ², where ρ is the sign imbalance. Looking at X only after a failure therefore selects a shifted distribution.
Failure rate–
corr(X, R), all runs–
E[X | failure]–
5% quantile of X | failure–
Failure agrees with {R ≤ 0}–finite-size mismatch → 0
EMeasurement against score
succeededfailedlimit E[R/σk | X]
Each dot is one graph. Height is the precritical score R/σk, read just before t*; failure ≈ {R ≤ 0}, the region below the gray line. The cloud tilts with slope ηkρ². With a tilt, cutting off the bottom also moves the X-coordinates of what remains. At ρ = 0 the cloud is round and the cut does not move X.
FDistribution of X, all runs vs failed runs
all runsfailed runsN(0, 1)selection law
Over all runs X is close to N(0, 1) whatever the signs. Over failed runs it follows the Gaussian-selection law Φ2(x, z; ηρ²)/Φ(z), drawn at the finite-size coordinate zsh = z₀ + CkM−1/6.
GConditional mean shift against ρ
−ηkρ²A(zsh), current kother ksimulated ± 2 s.e.
A(z) = φ(z)/Φ(z). The shift grows like ρ²: halving the imbalance cuts it by four. The k = 3, 4 curves rise and the k ≥ 5 curves fall, the sign reversal between k = 4 and k = 5. Every red point comes from the same graphs as the others; only the signs differ.
HLower tail of X given failure
empiricalΦ(x), unconditionalselection law
Capacity control reads a low quantile of X. On a log scale, a threshold set from the unconditional Φ has the wrong tail risk once failure has been observed. In the Gaussian limit with nonzero imbalance, failures push X up for k = 3, 4, making the unconditional rule conservative, and down for k ≥ 5, where it under-covers. At ρ = 0 there is no limiting shift. These are asymptotic directions; at finite M other errors, such as skewness of X, can partly offset or add to them.
Notes on Part II
Only the Plain model is simulated here, as in the main theorem. Signs are deterministic: the first ⌊d(1 + ρ)/2⌉ keys get +1. Keys are exchangeable, so this is equivalent to any fixed sign vector with that composition.
Measurement: T = Σv(Cv − kdρM/M)², d̂ = T/(k(1 − k/M)), X = (d̂ − d)/sM with sM² = 2d(d − 1)/(M − 1). It is taken from the initial sketch, before any peeling.
The score R is the backward-weighted functional of Part I, evaluated one step before t* or where peeling stops. Its agreement with actual failure improves with M, but it is not exact at finite size.
Predictions use the analytic constants with c = ηkρM². The shifted coordinate zsh is a finite-size diagnostic from the configuration setting; its transfer to Plain is not proved. The stat cards give the limit-coordinate value z₀ for comparison.