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• • • • • • ##### Activity $$f(x) = x^2 + y^2 bidyutsanki @bidyutsanki $x^2$ DivakaranDivakaran @DivakaranDivakaran x^2 $x^2$ Siddhartha Gadgil @siddhartha-gadgil Yes. It works. Not by marking it as code (as I tried) but using$$. DivakaranDivakaran
@DivakaranDivakaran
nice
any reference for the additivity i(a, c) + i(b, c) = i(ab, c) ? You mean to show it? If you a_ and _b are based geodesics, this is elementary.
Just count points. DivakaranDivakaran
@DivakaranDivakaran
ok. tha seems straight forward
that*
i had missed the assumption "based" The point is that based approximate actual
In the hyperbolic case, based are approximations with uniformly bounded additive errors. We need that usually geodesics are not almost parallel, as Moira, me and @arpan.into used arpaninto
@arpaninto
The paper of Chas and Lalley deals with the case when the surface has non-empty boundary. Does the result hold for closed case?
We can fix a standard generating set and ask the same question. Generally combinatorial group theory finds bounded case easier, but hyperbolic geometry should work for both. Just a gitter tip: ctrl-/ toggles chat and compose mode - compose mode is for many lines.
The interesting question here seems to be intersections for random geodesics. arpaninto
@arpaninto
So we have to following question: Given two random geodesics what is the limiting distribution of the intersection number when the limit is taken over the 1) hyperbolic length 2) word length? What do you mean precisely.
Is it a uniform distribution over words with length bounded above by some number. arpaninto
@arpaninto
Let $L$ be a positive number. Let $S_L$ be the curves with length (word or hyperbolic) less that $L$. Pick two geodesics randomly from $S_L$. Let the intersection number be $i_L$. What happens if $L\rightarrow infty$ ? We should try to formulate this in a general way.
We do need:
• A sequence of distributions so that the probability of picking geodesics of length below a fixed bound goes to 0.
• Choice is uniform in some appropriate sense.
I see the question.
There are interesting variants.
• We do have a completion of the set of geodesics (with multiplicity), namely the space of geodesic laminations.
• Perhaps we just need distributions that converge to a uniform distribution on these.
• I don't know exactly what uniform means here, but it should be satisfied by the limit of curves. It is probably best though to prove in one case. The methods will naturally generalize.
The natural starting case is uniform on geometric length $. arpaninto
@arpaninto
Lalley have done this count for self-intersection number. http://arxiv.org/abs/1111.2060
I was trying to read this paper but he proved them in general for negative curvature. The techniques involved there are completely new to me. Are these ergodic theoretic?
Ergodicity and mixing for geodesic flows are natural to use. But one may get something with direct geometry.
My impression of Lalley's work (from brief reading) is that he uses a little geometry and then squeezes a lot out of it using sophisticated probability.
Looking at the paper briefly, I do still feel that we should extract more from the geometric description of intersection numbers, in terms of lifts of curves, and use this. arpaninto
@arpaninto
Thank you Sir. I should probably concentrate on the first part (the computation of geometric intersection number for random curve of bounded length). Maybe random word length is more direct, but geometric length is similar.
Main idea:
• Pick an element of word (or geometric) length about $N\cdot L$.
• This is the sum of $N$ elements with word (or geometric) length about $L$.
• Further, these segments are themselves approximately independent and random.
• The intersection number is close to the sum of intersection numbers of pairs of these segments.
• We use a central limit theorem. arpaninto
@arpaninto
(I don't know how helpful this will be) By the following result of Dylan Thurston, It is enough to consider only (collection of) simple closed curves: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.112.8555&rep=rep1&type=pdf The ideas in that paper may be useful, even if the final result cannot be simply plugged in. arpaninto
@arpaninto
Try $\alpha$ Yes, \alpha\$ works inline too
Sorry, $\alpha$ arpaninto
@arpaninto
Sir about the third point of your main Idea: "Further, these segments are themselves approximately independent and random." Can you explain it a little bit? We can choose segments of length approximately $NL$ in two ways:
• We just choose among those of this length with equal probability, or
• independently choose $N$ segment of length approximately $L$ and take their concatenation
I claim that the two methods give approximately the same distribution
Of course everything gets complicated with lengths below a bound, rather than close to some fixed number.
Also, lengths are not additive - only approximately additive with high probability. arpaninto
@arpaninto
In Theorem 1.4 of Lalleys paper (http://arxiv.org/abs/1111.2060) he already computed the distribution of the geometric intersection number between two randomly chosen geodesics (bounded by geometric length). arpaninto
@arpaninto
It states that the proof is similar to the self-intersecting case. arpaninto
@arpaninto
He also states that "The methods of this paper can
be adapted to show that the main result of Chas extends to compact surfaces without
boundary and with genus $g \geq 2$."
$g\geq 2$ ritwik371
@ritwik371
Hi everyone, this is Ritwik. I would also like to join the discussion of discussing Moira Chas and Steve Lalley's result on the statistics of self intersection of loops. One idea I had is if we can prove a large deviation principle. I wrote up a few pages on that question; if you are interested, I can send you the write up.