Endowing numbers with meaning

It seems to be a quirk of the company I find myself in that my friends like setting me linear algebra teasers. My friend Dan introduced me to (sub)stochastic matrices this week, which are characterised by having nonnegative entries and row sums all (at most) 1. The incentive for this definition from probability is quite clear; one thinks of the entries as being the transition probabilities for some Markov process. He mentioned the following fact:

If $A$ is a substochastic matrix, then $0\leq\text{det}(\text{Id}-A)\leq 1$.

It’s quite a fun linear algebra exercise to prove this! I wrote a proof here alongside some elaboration on the rest of this post, but I suggest you try it yourself first. Pitman – who you’ve probably heard of from his probability book – preaches the mantra that any ‘interesting’ number between 0 and 1 should be regarded as a probability. The open problem associated to the above fact is…

What probability does $\text{det}(\text{Id}-A)$ describe?

It’s quite a probabilistic setup, so it’s not so outrageous to hope that some cute interpretation might exist.

If you’ve ever talked to me or skimmed my past posts, you will realise how distant this sort of question is from my usual research environment. However, I was quite intrigued because of the similarity of Pitman’s philosophy with an agenda of Kontsevich and Zagier pertaining to ‘periods’. A period is the integral of a rational polynomial over a region in $\mathbb{R}^n$ described by rational polynomial inequalities. Rational here means with coefficients in $\mathbb{Q}$… It’s quite apparent from the setup that periods are widespread in geometry; the name really comes from period integrals in Hodge theory. I give some examples in the note that’s linked above. The suggestion of Kontsevich and Zagier is that any ‘interesting’ transcendental number should be a period, and then that presenting such a number as a period (which of course has many different expressions as an algebraic integral) might explain apparently surprising equalities that crop up in geometry or number theory.