The x-axis represents the log10 of the gene locus length (defined by the locus definition.) The y-axis is the probability of a gene being assigned a peak. Each black dot on the plot is a bin of 25 genes, the y-axis value is then the proportion of peaks that were assigned to genes in that bin, and the x-axis value is the average of the locus lengths of genes in that bin. The spline curve is fit using a binomial smoothing spline model, see chipenrich
for more information. This curve is created by modeling presence of peak (a 0/1 binary variable denoting whether the gene was assigned a peak) given the gene locus length.
The random genes curve represents the model where each gene has the same probability of being assigned a peak given the total number of peaks in the dataset.
The "random peaks across genome" curve represents the model where the probability of a gene being assigned at least 1 peak is dependent on its length: $$p(peak\textbar L_{locus}) = 1 - (1 - \frac{L_{locus}}{L_{genome}})^{n_{peaks}}$$ $L_{locus}$ is the length of the gene locus, $L_{genome}$ is the length of the sampleable genome, and $n_{peaks}$ is the number of peaks in the dataset.