Geodesic
Random methods in 3-manifold theory
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, geometry, and number theory, Tome 292 (2016), pp. 124-148

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The surface map arising from a random walk on the mapping class group may be used as the gluing map for a Heegaard splitting, and the resulting 3-manifold is known as a random Heegaard splitting. We show that the splitting distance of random Heegaard splittings grows linearly in the length of the random walk, with an exponential decay estimate for the proportion with slower growth. We use this to obtain the limiting distribution of Casson invariants of random Heegaard splittings. 
@article{TM_2016_292_a7,
     author = {Alexander Lubotzky and Joseph Maher and Conan Wu},
     title = {Random methods in 3-manifold theory},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {124--148},
     publisher = {mathdoc},
     volume = {292},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_292_a7/}
}
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Alexander Lubotzky; Joseph Maher; Conan Wu. Random methods in 3-manifold theory. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, geometry, and number theory, Tome 292 (2016), pp. 124-148. http://geodesic.mathdoc.fr/item/TM_2016_292_a7/