Abstract. Let G be a finite group. We consider a connected graph such that the edges and
vertices are equipped with independent Poisson clocks (alarm clocks that ring at time dis-
tributed as the exponential distribution). Also, there are lamps with configurations indexed
by the elements of G and lamplighters at the vertices of the graph. The lamplighters at
a pair of neighbouring vertices exchange their position whenever the associated edge rings.
The lamplighter at a vertex updates the lamp configuration whenever the vertex rings. The
process can be viewed as a continuous-time random walk on the complete monomial group
GoSn. If the configuration of a lamp is x ∈ G, then it changes to g ·x ∈ G with rate αg (≥ 0).
We assume that αg = αg−1 for all g ∈ G, and the set {g ∈ G : αg > 0} generates G. We
show that the spectral gap of the process is the same as that of the lamplighter random
walk (i.e., the process with a single lamplighter) on the graph. This is an analogue of the
Aldous’ spectral gap conjecture for the complete monomial group of degree n over G.