  Masures (previously also known as hovels) are a generalization of the theory
of affine buildings for arbitrary $p$-adic Kac-Moody groups. Gaussent and
Rousseau invented masures to compute the Satake transform for $p$-adic
Kac-Moody groups. Their answer is given as a sum over Hecke paths, which are
certain piecewise linear paths.
  Guided by their method we give a definition of double affine Kazhdan-Lusztig
$R$-polynomials as a sum over piecewise linear paths that we call
$I_\infty$-Hecke paths. Remarkably, the notion of $I_\infty$-Hecke path, which
arises from masure theoretic considerations, is closely related to chains in
the double affine Bruhat order. Our main result is that there are finitely many
$I_\infty$-Hecke paths in untwisted affine ADE type. This implies that
$R$-polynomials are well-defined in this case. This finiteness result follows
from earlier known finiteness results for the double affine Bruhat order.
Combined with other results on the double affine Bruhat order, we now have all
the ingredients to define double affine Kazhdan-Lusztig $P$-polynomials.
