  We consider probability measures arising from the Cauchy summation identity
for the LLT (Lascoux--Leclerc--Thibon) symmetric polynomials of rank $n \geq
1$. We study the asymptotic behaviour of these measures as one of the two sets
of polynomials in the Cauchy identity stays fixed, while the other one grows to
infinity. At $n=1$, this corresponds to an analogous limit of the Schur
process, which is known to be given by the Gaussian Unitary Ensemble (GUE)
corners process.
  Our main result states that, for $n>1$, our measures asymptotically split
into two parts: a continuous one and a discrete one. The continuous part is a
product of $n$ GUE corners processes; the discrete part is an explicit finite
distribution on interlacing $n$-colourings of $n$ interlacing triangles, which
has weights that are rational functions in the LLT parameter $q$. The latter
distribution has a number of interesting (partly conjectural) combinatorial
properties, such as $q$-nonnegativity and enumerative phenomena underlying its
support.
  Our main tools are two different representations of the LLT polynomials, one
as partition functions of a fermionic lattice model of rank $n$, and the other
as finite-dimensional contour integrals, which were recently obtained in
arXiv:2012.02376, arXiv:2101.01605.
