Finite-Size Scaling and Data Collapse

Symbol Table

Symbol	Meaning
	Loop size (observable whose distribution is in question)
	System size (number of beads / chain length)
	Loop size distribution at system size
	Complementary CDF (survival function),
	Power-law exponent of the bulk (scale-free) region
	Cutoff exponent,
	Characteristic cutoff scale,
	Scaling function (cutoff function) of
	Scaling function for the CCDF
	Loop density,
	Moment order
	Marginal moment order,

Overview

“Is this distribution a power law?” is not a well-posed statistical question. At large sample size any pure power law is rejected by a KS test, while lognormal is nearly indistinguishable from a power law over a finite range. Finite-size scaling replaces it with a falsifiable question: does a single characteristic scale, set by system size, control all finite-size effects?

The scaling hypothesis is

The content of (1) is not “there is a power law.” It is:

When changes, the only thing that changes about the distribution is that one length grows. The curve does not change shape — it translates.

This is why the hypothesis is testable. Either curves at different collapse onto a single master curve after rescaling, or they do not.

Key Points

Structure of the scaling form

• — the scale-free part. No characteristic size.
• — the cutoff function. Depends only on the dimensionless ratio , never on and separately.

Asymptotics of are all that is assumed:

So for a pure power law is observed, and for the tail is truncated. marks where the power law stops being valid.

Note that is not an unknown to be tolerated — it is a measured output. The collapsed master curve is itself.

Physical meaning of

answers: how fast does the largest loop grow when the system grows?

	Interpretation
	Largest loops scale with chain length — extensive, loops occupy a finite fraction of the chain
	Largest loops grow slower than — sub-extensive, loops are a vanishing fraction as

The trivial bound guarantees ; the question is whether equality holds. In chromatin terms: does loop extrusion generate structure at the scale of the whole chromosome, or is there an intrinsic loop scale that does not grow with chain length?

This is the random-loop analogue of the crossover question already raised for the constant-loop case (whether ), which is a statement of the same type with .

Self-consistency of normalization

For the sum is dominated by small , converges independently of , and (1) needs no -dependent prefactor. For the cutoff dominates the normalization and an amplitude must be reinstated. Since the expected value here is , the form as written is consistent.

Testing collapse in practice

Treat as two fitting parameters and minimize a collapse residual:

where is a spline through the pooled rescaled points.

Two practical rules:

1. Collapse the CCDF, not the PDF. This eliminates binning freedom entirely and reduces noise. The exponent simply shifts by one:

Derivation: substituting in gives , so the same and the same appear with .

2. Plot contours of . If the minimum is a long valley rather than a sharp basin, and are degenerate along a compensating direction and neither can be quoted separately. Reporting a single minimum without checking this is a common error.

Moment scaling — an independent route to and

Moments give access to the same exponents without any tail fitting:

Small is bulk-dominated and -independent; large is cutoff-dominated and grows as a power of . The crossover sits at .

For the first moment is exactly marginal, giving a sharp diagnostic:

So plotting against and checking for a straight line tests directly, and the slope gives . This is far more stable than tail fitting: the mean uses the whole dataset, so statistical error is small, and there is no arbitrary choice. This is the cheapest high-value measurement available.

Failure modes

Observation	Interpretation
No collapse for any	More than one characteristic scale, or a crossover
drifts with	Apparent power law; lognormal or similar
depends on beyond	Wrong scaling variable — see below
valley degenerate	Exponents not separable; widen the range of

The density trap

What is held fixed while is scanned must be decided explicitly. Holding the loop number fixed (e.g. ) while increasing changes the loop density simultaneously. Then observed changes cannot be attributed to , and the true scaling form is a two-variable function:

Single-variable collapse then fails in principle, not because the hypothesis is wrong. Fix and scan . If is itself of interest, establish the collapse at fixed first, then study dependence as a second stage.

Questions & Insights

Related Concepts

• 2025-05-30_Ising_Model_Finite_Size_Scaling
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• 🕯️ Infinite step-rate limit of symmetric loop extrusion (0729)

Notes from Claude