### 3.5 Non-systematic dispersion

As mentioned in Section 2.6, Lorentz violation is only one possibility for a signal of quantum gravity.
Another common idea about quantum gravity is that spacetime should have a stochastic [99, 255] or
“foamy” [231] structure at very small scales. Combining these two ideas has lead a number of authors to
the idea of stochastic/non-systematic dispersion where the modifications to the dispersion relation fluctuate
over time. Such dispersion modifications have been phenomenologically parameterized by three numbers,
the usual coefficient and exponent of Section 3.1, and a length scale which
determines the length over which the dispersion is roughly constant. After a particle has travelled a
distance a new coefficient is chosen based upon some model dependent probability
distribution that reflects the underlying stochasticity. Usually is assumed to be a gaussian
about , such that the average energy of the particle is given by its Lorentz invariant value.
As well, is in general taken to be the de Broglie wavelength of the particle in question.
Note that in these models is not required to be an integer, the most common choices are
[232].
The only existing constraints on non-systematic dispersion come from threshold reactions (see
Section 6.6.2) and the phase coherence of light (see Section 6.9).