Exact relations between grouped three-dimensional intersection exponents

Determine whether the apparent equalities between the three-dimensional intersection exponents for configurations $(1,4)$ and $(2,2)$, and for configurations $(2,4)$ and $(1,1,3)$, are exact relations arising from corresponding non-intersection constraints.

Background

The paper numerically compares intersection exponents for different ways of grouping simple random walks in three dimensions. For four walks, the estimates for $(1,4)$ and $(2,2)$ differ by only $0.0013(14)$, while for five walks, the estimates for $(2,4)$ and $(1,1,3)$ differ by $0.01(2)$; both differences are statistically consistent with zero. These close numerical values suggest that distinct grouping arrangements may impose geometrically related avoidance constraints.

The unresolved issue is whether either numerical similarity reflects an exact identity between the corresponding intersection exponents, rather than an agreement caused by finite-size effects or statistical uncertainty. Establishing such relations would clarify the structure of the three-dimensional intersection-exponent spectrum.