A Rayleigh matroid is a matroid with set of matroid bases 
whose basis generating polynomial 
satisfies the Rayleigh inequalities
 |
for every pair of distinct ground set elements 
and all positive weights 
(Choe and Wagner 2006). Here 
denotes partial
differentiation with respect to 
. The condition expresses pairwise
negative correlation in the probability measure on matroid
bases proportional to 
.
The stronger 
-Rayleigh
property uses
 |
where 
is the matroid rank, and requires the same derivative
inequality for every 
. Zhao and Zhou (2026)
proved that every finite matroid
of rank or corank at most 3 is 
-Rayleigh. Such matroids also satisfy
the Rayleigh inequalities for the generating polynomials of independent sets and
spanning sets. This strengthens the previously known rank-three basis-Rayleigh theorem.
The rank restriction cannot be discarded: the rank-four matroid 
already fails the basis-Rayleigh property (Choe and Wagner
2006).
-Rayleigh." 11 Nov 2026.