On measure concentration in graph products

Bollobás and Leader [1] showed that among the n-fold products of connected graphs of order k the one with minimal t-boundary is the grid graph. Given any product graph G and a set A of its vertices that contains at least half of V (G), the number of vertices at a distance at least t from A decays (as t grows) at a rate dominated by P(X1 + . . . + Xn t) where Xi are some simple i.i.d. random variables. Bollobás and Leader used the moment generating function to get an exponentialbound for this probability.We insert a missing factor in the estimate by using a somewhat more subtle technique (cf. [3]).


Introduction and theorem
Consider a finite set [k] consisting of k elements: {0, 1, . . . , k − 1}. We may define various metrics (distances) d on [k]. One of the ways to do that is to consider a graph G = (V , E) with a vertex set V = [k] and define the distance d(a, b), as the length of the shortest path between a and b. In order to have a finite metric, we will, of course, put a restriction that the graph G is connected.
If, for example, we choose G to be a path P k , i.e., graph with the edge set E = {{0, 1}, {1, 2}, . . . , {k − 2, k − 1}} then the resulting metric is the one inherited from the real line with the Euclidean distance. On the other hand, if G is a complete graph K k on k vertices, consisting of all possible pairs of vertices, then d(a, b) = 1 iff a = b.
Let us consider a product [k] n of metric spaces ([k], d 1 ), . . . , ([k], d n ) each with the same number of elements but probably distinct metrics d i . Let us denote elements of [k] n as a = (a 1 , . . . , a n ).
It is easy to see that the l 1 -type metric on [k] n defined as is indeed a metric. We choose this way of defining a metric on the product space because we can reconstruct a graph on [k] n by considering a pair {a, b} an edge if and only if d(a, b) = 1. If metrics d i are induced by graphs G i we shall refer to the graph reconstructed from the metric d as the cartesian product of graphs G i , i = 1, . . . , n, denoting it G = G 1 × . . . × G n . We can equivalently define G by saying that a pair {a, b} of vertices is an edge whenever there is i such that {a i , b i } is an edge in G i and a j = b j for all j = i.
Consider the example where G i = P k . Multiplying a path by itself we obtain so called n-dimensional grid graphs.
Given a subset of vertices A ⊂ V of a graph G which is not too small (say, has at least |V |/2 elements), how big is its neighbourhood, i.e., vertices having a neighbour in the set A? More generally, how many vertices are there at a distance from A at most t?
Let us denote t-neighbourhood of A as A t := {a ∈ V : d(a, b) t for some b ∈ A}. Given a graph, we are interested in finding a set that has the smallest t-boundary, it is, determining the quantity It turns out that in the case of product graphs of high dimension a striking phenomenon (known as concentration of measure) is observed: A t is almost all of V whenever t is a small proportion of the diameter of G.
We may pose a question from another point of view: given a class of graphs, which one has the slowest growth of A t , or, seeking a slightly weaker answer, what is a good lower bound for (1)? This was fully answered by Bollobás and Leader [1] in case when the class consists of all n-fold products of graphs on k vertices.
Consider, for r 0, balls around zero B THEOREM 1 [Bollobás and Leader, [1]]. Let G 1 , . . . , G n be connected graphs of The lower bound given by Theorem 1 can be interpreted using probability. Let X 1 , . . . , X n be independent copies of a random variable X distributed uniformly over [k]: Now we can estimate |B (n) k (r + t)| by the means of the following representation: Let Bollobás and Leader [1] estimated the moment generating function exp{hS n } by calculating moments of S n and then used Chebyshev's inequality to obtain the following statement.
THEOREM 2 [Bollobás and Leader [1]]. Let G 1 , . . . , G n be connected graphs of order k and let where S n is the random variable defined in (4) and nσ 2 = VarS n .
Using the Central Limit Theorem we can see that the constant 6/(k 2 − 1) in (6) cannot be improved. However, one could expect a bound similar to the right tail of a normal random variable with variance nσ 2 . We show that this is indeed the case. THEOREM 3. For the random variable S n defined in (4) and t ∈ R we have Theorem 3 gives an improvement upon the bound (6) whenever t is of order larger than σ √ n which is the case when we set t to be a 'small fixed proportion' of the diameter of the grid graph, namely t = ε diam(P n k ) = εn(k − 1). with an arbitrarily small ε > 0.

Proof of Theorem 3. Consider, for any h < t, a function
Applying Lemma 3 and Lemma 1.1 of [2] we conclude the proof.

Lemmas and their proofs
Consider a random variable τ = τ (b, σ 2 ) which assumes values {−b, 0, b}, with probabilities where Y is a centered discrete uniform random variable on [k] as defined in (4) and Proof. Note that Y is symmetric and so satisfies the conditions of Lemma 3 of [5] with b = max Y and σ 2 = VarY . Therefore we get that for h ∈ R To prove (8) it suffices to show that + has the second continuous derivative. Therefore we can differentiate f under the integral to obtain It is obvious that The following result is probably the essence of the paper.
where η is a normal random variable with mean zero and variance σ 2 .
Proof. For simplicity and without loss of generality we may assume that b = 1, because the general case follows by rescaling. Under this assumption we have that σ 2 1/3. To prove the lemma it suffices to show that E(η − h) 5 = 5h − 3σ 4 + 2 · 1 4 · σ 2 /2 0 since σ 2 1/3 and odd moments of symmetric random variables vanish. Case 3. h ∈ [−1, 0]. We may reduce this case to the Case 4 as soon as we show that It is easy to check that function f restricted to the interval [0, 1] is five times differentiable and its k-th derivative is where c k are positive constants, and we make a convention 0 0 = 1.
The following argument is clear if one looks at the graphs of f (k) . Note that f (5) (5) is increasing. By Chebyshev's inequality we have P(η > 1) σ 2 /2, so f (5) (1) 0. Consequently, there is a number x ∈ [0, 1] such that f (5) 0 on [0, x] and f (5) In order to see how the sign of f (3) varies, we observe that In order to see how the sign of f varies we check that LEMMA 3. Let the random variable S n be defined as in 4. Then for any h < t we have where η is a centered normal random variable such that VarZ n = VarS n = nσ 2 .