Exact Hausdorff gauges for SLE

The explanation

To measure a smooth curve, cover it with small discs and add up their sizes. A curve of length 1 needs about 1/r discs of width r. Their widths add up to about 1, whatever r is. The power of r that makes the total settle down is the curve's dimension, here 1.

Some random curves are so wiggly that this fails. Schramm–Loewner evolution, or SLE, is a family of random curves in the plane. They describe boundaries in critical models of physics, such as percolation clusters. A parameter κ sets how wiggly the curve is. For κ between 0 and 8, Beffara showed that the dimension is d = 1 + κ/8Source 1. So with κ = 4, d = 1.5.

One might then measure the curve with r^1.5 in place of r. Rezaei showed that this gives zeroSource 2. The pure power is slightly too small. It needs a tiny correction factor, and the full size rule is called a gauge. Schramm asked for a gauge that gives the curve a sensible, nonzero sizeSource 3. He suggested r^d × log log(1/r).

The first paper claims that a suitable gauge exists for every κ between 0 and 8Source 4. It gives every piece of the curve a positive, finite size. But the gauge is defined through integrals, and its shape need not be a neat formulaSource 5.

The second paper names the gauge outright. It is h(r) = r^d × (log log(1/r))^((2−d)/2). It claims this gives every positive-time piece of the curve a positive, finite sizeSource 6. The paper calls this an explicit solution to Schramm's problemSource 7. The whole curve inside any bounded disc also has finite expected sizeSource 8.

Try κ = 4 and r = 10^(−6). Then log(1/r) is about 13.8, and its log is about 2.63. The correction is 2.63^0.25, about 1.27. Schramm's suggestion uses the full 2.63. The ratio of the two corrections is 2.63^(−0.75), about 0.48. At r = 10^(−100), it falls to about 0.28, and it keeps shrinking. So Schramm's gauge grows too fast in the limit. The paper claims it gives no sensible size on any piece of the curveSource 9.

The obstacle is that both directions need fine control at every scale. A lower bound needs a mass spread on the curve that is never too heavy in any small discSource 10. An upper bound needs covers whose total cost stays small on averageSource 11. For the lower bound, the paper deletes small squares that hold too much mass. The cost of reaching a square, its mass, and the number of squares balance out exactlySource 12. With κ = 4, the exponents 0.5 and 1.5 add to 2, the dimension of the plane.

Other ways to measure SLE exist. The natural parametrization measures the curve along its own path. Holden and Yuan found a different log correction for ordered pieces of the curve. The gauge here belongs to arbitrary covers insteadSource 13. The argument needs no link to those other measuresSource 14. How the two exact gauges relate is still openSource 15.

How sure is this? Lean, a computer proof checker, has a formal version of one direction. For the explicit gauge, it states that every piece of the curve gets positive size, almost surely. The finite upper bound and the first paper's gauge are not in the scope note. No specialist review of these papers has been published.

What it doesn't show. This does not say how the explicit gauge relates to the first paper's gauge.