Very good. I am neither a scientist nor a mathematician, but I do read a bit of the literature occasionally and tend to think about it. You have addressed one of the points that has worried me for a little while. For example, I sent the following email to Erich Fischer in December 2013:
Last August I read a theoretical paper (attached) that caused me to think seriously about alternatives to ensemble means (whether multimember or multimodel) in considering possible climate change consequences. Therefore, I was very interested when a blogger referred me to your 13 Nov NCC paper with Beyerle and Knutti. I realize that the type of model considered in the attached paper is (infinitely) simpler than the climate models you are working with. Nevertheless, I wondered if something similar to the theoretical conclusion might not hold for much more complex numerical models. And I also wondered whether it is possible to construct useful measures from the ensembles if we have reason to expect that the ensemble mean is not a good indicator of what might be expected. Your paper is a strong indicator of one such useful alternative measure. “The perspective is more informative than a global mean and when taking into account local vulnerabilities it may be promising in many fields, such as the reinsurance business with globally distributed portfolios, the global commodity market or strategies for global food security.”
Thinking in very general terms, I expect that the critical threshold values for local systems are not constant but depend on the system “resilience.” Further, I suspect that it is not simply a question of exceeding the critical threshold. Instead, I think we may have a probability of cascade into a different system phase as a time-measure integral of a function of the above-threshold magnitude. If this is so, then being able to estimate the likelihood of different time patterns of above-threshold values might be useful.
Anyway, thank you for your stimulating paper.
The NCC paper referred to is http://www.nature.com/nclimate/journal/v3/n12/full/nclimate2051.html
The paper I attached to the email is V. I. Klyatskin, “Clustering of a positive random field as a law of nature,” Theor. Math. Phys., 176(3): 1252–1266 (2013).