Intrinsic and extrinsic geometry of random surfaces

TL;DR

This paper investigates the geometric properties of triangulated random surfaces, establishing inequalities between intrinsic and extrinsic dimensions and exploring their relation to critical phenomena.

Contribution

It proves that extrinsic Hausdorff dimension exceeds or equals intrinsic dimension in certain models and derives scaling relations linking intrinsic dimension to critical exponents.

Findings

Extrinsic Hausdorff dimension ≥ intrinsic Hausdorff dimension

Intrinsic Hausdorff dimension may be infinite at critical points

Scaling relations connect intrinsic dimension to critical exponents

Abstract

We prove that the extrinsic Hausdorff dimension is always greater than or equal to the intrinsic Hausdorff dimension in models of triangulated random surfaces with action which is quadratic in the separation of vertices. We furthermore derive a few naive scaling relations which relate the intrinsic Hausdorff dimension to other critical exponents. These relations suggest that the intrinsic Hausdorff dimension is infinite if the susceptibility does not diverge at the critical point.