BEST statistics of Markovian fluxes: a tale of Eulerian tours and Fermionic ghosts

New paper out!

MP, BEST statistics of Markovian fluxes: a tale of Eulerian tours and Fermionic ghosts,
J. Phys. A: Math. Theor. 48, 365005 (2015), arXiv:1503.03045

This paper represents my first attempt of an incursion in the field of Large Deviation Theory. I like it because it connects results in two different areas, namely graph theory and probability theory (sneaking into the territories of Quantum Field Theory), though I understand that it is a rather obscure paper that will find few readers.

In this paper I prove that

formula

OK I just wanted to show off a long formula that means nothing. So suppose a person is blindfolded, cast away in the middle of the desert and asked to walk in some random direction. How much sweat and tears will he pour? Of course, the answer to the above question is probabilistic, since every time this strange experiment is repeated the person (assuming he is still alive and memoryless) will walk a different path, due to the influence of environmental disturbances, sand, wind, sun, scoprions etc. According to the path he follows perspiration and desperation will vary.

In mathematical terms, this person is performing a random walk, and the question we are asking is to calculate some number characterizing how “expensive” (in thermodynamic terms) the trajectory is.

So, quite unremarkably, it turns out that… the answer does not exist. In fact, unfortunately, assuming we know how much sweat and tears it takes to walk every possible step in the desert, we still don’t know which exact sequences of steps produce exactly N liters of sweat and tears.

Probability theory does not have the general answer to this question, but does it have an answer to any question at all?

Indeed, yes. If, instead of focusing on sweat and tears, we ask the much more detailed question: for every possible step in the desert, how often did the walker walk that step?, then we have the answer. That’s what my paper is about. Before my paper, answers could be provided after the walker walks for an infinitely long time. My work extends this before infinity. More details might be coming, or not.

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