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    Siddhartha Gadgil
    @siddhartha-gadgil
    It is probably best though to prove in one case. The methods will naturally generalize.
    The natural starting case is uniform on geometric length <L<L.
    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.
    Siddhartha Gadgil
    @siddhartha-gadgil
    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).
    Siddhartha Gadgil
    @siddhartha-gadgil
    Maybe random word length is more direct, but geometric length is similar.
    Main idea:
    • Pick an element of word (or geometric) length about NLN\cdot L.
    • This is the sum of NN elements with word (or geometric) length about LL.
    • 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
    Siddhartha Gadgil
    @siddhartha-gadgil
    The ideas in that paper may be useful, even if the final result cannot be simply plugged in.
    arpaninto
    @arpaninto
    Try α\alpha
    Siddhartha Gadgil
    @siddhartha-gadgil
    Yes, $$\alpha$ works inline too
    Sorry, α\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?
    Siddhartha Gadgil
    @siddhartha-gadgil
    We can choose segments of length approximately NLNL in two ways:
    • We just choose among those of this length with equal probability, or
    • independently choose NN segment of length approximately LL 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 g2g \geq 2."
    g2g\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.
    arpaninto
    @arpaninto
    Hi Ritwik, I am arpan. Can you send the write-up to me. My email id is : arpan.into@gmail.com
    Siddhartha Gadgil
    @siddhartha-gadgil
    I am here: for convergence of laminations.
    DivakaranDivakaran
    @DivakaranDivakaran
    Hello sir
    I fear taking punctures as in gromov's proof might not work. Not having a mattulla tyoe lemma, will be the main obstacle
    Margolis type lemma*
    Sorry
    DivakaranDivakaran
    @DivakaranDivakaran
    Margulis type lemma*
    Siddhartha Gadgil
    @siddhartha-gadgil
    As I mentioned in my mail:
    • start with the principle that nothing comes free.
    • so there must be an example where a compactness result for laminations holds in a non-trivial way.
    • in this case, non-trivial means that a limit of surfaces as a surface does not exist, but it exists as a lamination.
    • this can happen if (and I suspect only if) locally the maps are well-behaved, but the domains have genus going to infinity.
    • given one such example, it makes sense to try to prove a theorem around it.
    DivakaranDivakaran
    @DivakaranDivakaran
    Yeah, makes sense. I will think about it
    Siddhartha Gadgil
    @siddhartha-gadgil

    Quoth Dheeraj:

    I went through the gitter conversation. I agree that it is essential to come up with a protypical example where J-holomorphic curves converge to a genuine lamination (Else, as you point out, there is nothing much gained by passing to laminations).

    Now I am thinking of examples in CP^2 (in general CP^n) to see if there is such an example. I am looking at curves with varying genera so that one avoids bubbled curves. I will post it if I see something new.

    The one candidate example I suggested some years ago was Donaldson's construction, taking limits of approximately holomorphic curves as the approximation constant goes to 0.
    Another possibility is if, given a J-holomorphic curve, we can view this as the image of a cover, and then straighten out to reduce second derivative. I have no idea when or why straightening is possible.
    The most intriguing one is whether a Kahn-Markovic style construction gives something in the limit.
    DivakaranDivakaran
    @DivakaranDivakaran
    Sullivan had suggested the second possibility - considering a limit of subsequent covers- to me years ago
    We will think along these lines
    DheerajKulkarni
    @DheerajKulkarni
    The one candidate example I suggested some years ago was Donaldson's construction, taking limits of approximately holomorphic curves as the approximation constant goes to 0.
    Yes.. will look into it
    Donaldson's construction with CP^2 as base might be easier to handle.