Global food prices are typically analysed in a time-series framework. We complement this approach by focusing on the spatial price dispersion of the country-pair bilateral trade in the international food trade network (), for ten relevant commodities. The main purposes are to verify if the Law of One Price () holds and to investigate the emergence of randomness in the price-formation mechanism. We distinguish between the "internal" variance, which indicates the magnitude of price discrimination, and the "external" variance, that is a measure of price dispersion. We find that, for some commodities, spatial price dispersion is remarkable and persistent over time (i.e., failure of the ) and that there exists a strict correlation between price spikes and peaks in spatial price variability. We test whether the price distribution can be replicated through a stochastic process of extraction. Surprisingly, the actual distribution of prices, for several commodities, is well described by a random distribution. Then, the process of data aggregation is not neutral because the information at the micro-level scale might be lost at the macro-scale, due to the complexity of the . Finally, we discuss some possible economic explanations of these outcomes and the main methodological, environmental, and policy consequences.