On monochromatic arm exponents for 2D critical percolation

TL;DR

This paper studies monochromatic arm exponents in 2D critical percolation, showing they form a distinct family with specific bounds between known polychromatic exponents, advancing understanding of percolation probabilities.

Contribution

It establishes the existence and bounds of monochromatic arm exponents, revealing they differ from polychromatic exponents and are strictly between them.

Findings

Monochromatic j-arm exponents exist in 2D critical percolation.

These exponents form a new family distinct from polychromatic exponents.

Monochromatic j-arm exponents are strictly between polychromatic j-arm and (j+1)-arm exponents.

Abstract

We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j+1)-arm exponents.