"This week, we examined the simplest theory of spatial criminology – that crimes are distributed at random. When spatial criminologists want to test observed data for randomness, they test that data’s distribution against the Poisson distribution and look for a statistically significant difference. What is the null hypothesis in this situation? What conclusion would we draw from rejecting the null hypothesis?"
If the occurrence of events is homogeneous and independent then the number of events per unit of exposure (e.g. area) will have a Poisson distribution, so you could test for a null of that particular kind of randomness by doing a goodness of fit test of a Poisson distribution.
You can then phrase the null either as a null of the process being random (a Poisson process) or as having a Poisson distribution.
If your null is of a non-homogeneous process then things become more complicated than that.
I would add that this seems a highly implausible model and I see no good reason to be testing for it.
If you want to evaluate the pattern of dispersion of points in space (such as the distribution of crimes in some area), a homogeneous Poisson process (a.k.a. "Complete Spatial Randomness" or CSR) is a very useful dividing hypothesis. The standard way to compare the observed pattern to the expected pattern under CSR is with some sort of summary function, such as Ripley's K-function. The idea is to compare the function for the observed data to the expected function under CSR; if the pattern deviates from CSR in a particular direction, it might indicate that the pattern is more clumped/clustered than expected under CSR...or, conversely, that the pattern is more uniform/regular than expected under CSR. If we want to take it a step further, we can then evaluate other point-process models that might be more appropriate. Beware, however, that if the observed pattern deviates from CSR in the direction of clustering, it might indicate a cluster process or it could be due to inhomogeneity (see the link below).
If you want to test against the null hypothesis of CSR, the standard approach is to simulate multiple realizations of CSR to build a significance envelope (which tells us something about the range in the behavior of the summary function that we might observe under the null hypothesis). Because second-order functions such as Ripley's K allow you to evaluate the pattern of dispersion at varying distances, there are different forms of envelopes that can be constructed depending on whether you have an a priori hypothesis about the pattern of dispersion at a particular distance between points...or if you want to test whether the observed function deviates from the expectation under CSR at any distance. Such tests can be one or two tailed.
If you are interested in learning more about analyzing patterns of dispersion, I highly recommend Baddeley et al. (2015), which is associated with the 'spatstat' package in R.
You might also find this blog post useful, wherein I discuss visualizing and accounting for inhomogeneity (when assessing dispersion with a summary function). The post contains some example applications of Besag's L-function (which is the variance stabilized transformation of Ripley's K-function), illustrating some of the concepts discussed above.