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The questions from the probabilistic combinatorics section of the mathematical tripos part iii exam held on june 7, 2010. The questions cover topics such as random hypergraphs, concentration of probability inequalities, and talagrand distance. Students are required to attempt no more than three questions out of the four provided, and the questions carry equal weight.
Typology: Exams
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Monday, 7 June, 2010 9:00 am to 11:00 am
Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
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(i) Let G (^) N(r,p) be the random r-uniform hypergraph with vertex set N in which every r-set
is chosen with probability p, where 0 < p < 1. Show that a.s. G (^) N(r,p) is isomorphic to a
fixed r-uniform hypergraph G (^) univ(r).
(ii) Delete the maximal vertex of each hyperedge E of G (^) univ(3) from E to obtain a graph G 0.
Show that G 0 is isomorphic to G (^) univ(2).
(i) Prove a concentration of probability inequality for a Lipschitz function on the symmetric group Sn, considered as a probability space, with Hamming distance.
(ii) Let A and B be non-empty subsets of Sn, and write d(A, B) for their Hamming distance. Show that min{|A|, |B|} 6 n! e−d (^2) / 8 n .
[The results you use should be clearly stated.]
Let z = (z 1 ,... , zn) be a sequence of n points chosen at random in the right-angled triangle with vertices (0, 0), (1, 0) and (1, 1), and let Ln = Ln(z) be the maximal number of points zi forming a convex polygon with (0, 0) and (1, 1). [Thus Ln(z) is the maximal ℓ such that the points (0, 0), zi 1 , zi 2 ,... , ziℓ and (1, 1) are the vertices of a convex (ℓ + 2)- gon.]
(i) Sketch a proof of the fact that the median of the random variable Ln is at most cn^1 /^3 for some c > 0. [You may find it helpful to consider the triangles formed by the tangents of the hyperbola y = x^2 at the points (k/n^1 /^3 , k^2 /n^2 /^3 ).]
(ii) Let 1 6 b < a be positive integers, and set A = {z : Ln(z) > a} and B = {z : Ln(z) 6 b}. Show that
dT (A, B) >
a − b √ a
where dT (A, B) is the Talagrand distance of the sets A and B.
(iii) Let ω(n) → ∞. Prove that whp Ln is concentrated in an interval of length ω(n)n^1 /^6.
Part III, Paper 12
