Exact Hausdorff gauges for SLE

1 · The essentials

How can we assign a meaningful size to a complicated random curve? These manuscripts address that question for a family of random curves called chordal SLE, with a parameter $0 < κ < 8$ . The goal is a measuring rule that gives each nontrivial piece of the curve a size that is neither zero nor infinite. The first manuscript supplies such a rule, chosen independently of the random curve. The companion manuscript gives an explicit formula for it. With probability one, the rule works for every piece traced between two finite, strictly positive times. It also gives a finite average measure for the part of the whole curve inside any bounded disk. The main result has a Lean formalization, although the supplied source does not specify which individual claims that formalization covers.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine measuring a tangled thread with small patches, each carrying a price determined by its width. Choosing the right pricing rule makes the total cost stay meaningful as you use smaller patches; here, the manuscripts identify such a rule for the random curves under consideration.

Why it matters

The manuscripts describe these results as answering Schramm’s Hausdorff-measure existence problem for chordal SLE in the range $0 < κ < 8$ . The explicit formula turns an existence statement into a specified measuring rule. The companion also distinguishes this rule from the exponent-one iterated-logarithm gauge suggested by Schramm, which it reports is not sigma-finite on the relevant curve segments.

2 · Going further

The problem is to find a deterministic Hausdorff gauge for chordal Schramm–Loewner evolution, or SLE, that assigns positive finite measure to its nontrivial compact positive-time trace segments. For each fixed $0 < κ < 8$ , the desired statement holds almost surely for every segment $γ ([s, t])$ with $0 < s < t < \infty$ . The source identifies this as Schramm’s Hausdorff-measure existence problem in that parameter range.

The main manuscript constructs a gauge with this property and states that the entire trace has finite expected gauge measure in every bounded box. The companion makes the gauge explicit: with $d = 1 + κ /8$ , it uses $h (r) = r^{d} (lo g lo g (1/ r))^{(2 - d) /2}$ for sufficiently small radii $r$ . It states the same positive-finite segment conclusion and finite expected measure for the whole trace restricted to any bounded disk.

The construction described at the abstract level is therefore to choose a deterministic scale-dependent measuring rule, with the companion identifying a power of the radius corrected by an iterated logarithm. The abstracts do not supply the argument establishing the measure bounds, so no more detailed proof mechanism can be extracted from this source. The companion further states that Schramm’s suggested exponent-one iterated-logarithm gauge is not sigma-finite on any such segment. The main result has a Lean formalization, but the source does not delineate its coverage of the companion’s formula or additional claims.

Useful notions

Random curves and probability-one statements
Measuring sets with small covers
Hausdorff measures and gauge functions
Powers and iterated logarithms

What was known before / what is new

Before

The source identifies a Hausdorff-measure existence problem posed by Schramm for SLE. It also mentions an exponent-one iterated-logarithm gauge suggested by Schramm. No further account of earlier results or methods is supplied.

New

The main manuscript shows the existence of a deterministic gauge giving almost surely positive finite measure to every nontrivial compact positive-time segment in the stated parameter range. The companion gives the explicit gauge $r^{d} (lo g lo g (1/ r))^{(2 - d) /2}$ and states finite expected measure in every bounded disk. It also reports that the exponent-one iterated-logarithm gauge is not sigma-finite on these segments.

Limits and assumptions

The results concern chordal SLE with a fixed parameter strictly between 0 and 8. The segment statement requires $0 < s < t < \infty$ , and the explicit gauge formula is specified only at sufficiently small radii. The finite-expectation statements are local, concerning bounded boxes or disks, not the unrestricted whole trace. The main result has a Lean formalization; the supplied source does not specify its claim-by-claim scope, so formal verification of every companion claim should not be inferred. The source states no conclusion at the parameter endpoints or outside this range and does not list further open problems.

Mini glossary

Chordal SLE: Chordal Schramm–Loewner evolution is the parameterized family of random curves studied in these manuscripts.
Trace segment: A trace segment $γ ([s, t])$ is the part of a curve visited between times $s$ and $t$ .
Hausdorff gauge: A Hausdorff gauge is a rule assigning a scale-dependent weight to the small covering sets used to measure a set.
Almost surely: A statement holds almost surely when it holds with probability one.
Iterated logarithm: The iterated logarithm in this formula is $lo g lo g (1/ r)$ , obtained by applying the logarithm twice.
Sigma-finite: A measure is sigma-finite on a set if that set can be covered by countably many subsets of finite measure.

Keywords : Chordal SLE · Hausdorff gauge · Iterated logarithm

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Resolves Schramm’s Hausdorff-measure question for chordal SLE_κ, $0 < κ < 8$ . The explicit gauge $r^{d} (lo g lo g (1/ r))^{(2 - d) /2}$ , $d = 1 + κ /8$ , gives almost surely positive finite measure to every trace segment $γ ([s, t])$ with $0 < s < t < \infty$ , and finite expected measure to the trace in every bounded disk.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/230.md

Manuscripts in this family

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An exact Hausdorff gauge for SLE

25/09/2026

To be verified

For each $0 < κ < 8$ , we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.

PDF GitHub folder

An explicit exact Hausdorff gauge for SLE

26/09/2026

To be verified

For each fixed $0 < κ < 8$ , let $d = 1 + κ /8$ . The gauge $h (r) = r^{d} (lo g lo g (1/ r))^{(2 - d) /2}$ at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLE_κ. This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.

PDF GitHub folder