An entropic proof of cutoff on Ramanujan graphs

It is recently proved by Lubetzky and Peres that the simple random walk on a Ramanujan graph exhibits a cutoff phenomenon, that is to say, the total variation distance of the random walk distribution from the uniform distribution drops abruptly from near $1$ to near $0$. There are already a few alternative proofs of this fact. In this note, we give yet another proof based on functional analysis and entropic consideration.

We consider the simple random walk (X t ) t∈N on G starting at some X 0 ∈ G. The probability distribution µ t := P t ( · , X 0 ) of X t converges to the uniform distribution π := 1 |G| 1 G . For α ∈ (0, 1), the total variation mixing time is defined to be (2) T mix (α) := min{t ∈ N : µ t − π TV < α}, is the total variation norm. The total variation mixing time T mix (α) is clearly monotone in α. Since the random walk (X t ) t backtracks with probability 1/d, the expected distance from the origin X 0 to X t is at most d−2 d t. Thus µ t is concentrated on the ball of radius . This observation leads to the entropic lower bound of the total variation mixing time, d is the asymptotic entropy of the simple random walk on the d-regular tree (see [LP, BL] for a more precise estimate). On the other hand, since µ t −π 1 ≤ |G| 1/2 µ t −π 2 ≤ |G| 1/2 ρ t , one has the spectral upper bound This upper bound also allows an entropic interpretation, see Lemma 2 below. Now, let G = {G} be a family of graphs. Recall that G is said to be a family of expanders (see e.g., [Lu]) if (7) sup{ρ G : G ∈ G} < 1.
When needed, we add the subscript G to the symbol ρ etc to specify the graph G under consideration. Note that the Alon-Boppana bound gives the lower estimate for any family G. Here the value ρ d is the spectral radius of the simple random walk on the d-regular tree. The family G is said to be asymptotically Ramanujan if it marks the Alon-Boppana bound, An explicit family of Ramanujan graphs was first discovered by Lubotzky-Phillips-Sarnak ( [LPS]) and it was shown by Friedman ( [Fr]) that the random graphs are asymptotically Ramanujan. We have seen in the above that every expander family satisfies (10) T mix G (α) ≍ α log |G| as |G| → ∞. The family G is said to exhibit cutoff if the implied constants in the above asymptotic equality are independent of α, or more precisely, if for every α, α ′ ∈ (0, 1) and every choice of the starting points X 0 G ∈ G. It is proved by Lubetzky and Peres ( [LP]) that asymptotically Ramanujan graphs exhibit cutoff.
Theorem 1. A family G of asymptotically Ramanujan graphs exhibits cutoff. In fact, the total variation mixing time marks the entropic lower bound: For any α ∈ (0, 1), See [He, BL] for alternative proofs of this theorem. In this note, we give yet another proof of it.

Entropy
Let Prob(G) denote the set of probability measures on a (finite or infinite) graph G. Recall that the Shannon entropy (or just entropy) of ν ∈ Prob(G) is the quantity When a random walk (X t ) t∈N on G is under consideration, we also write Let's assume for the moment that the graph G is transitive. Then the quantity H(t) is increasing and concave. The limit h := lim t 1 t H(t) = lim t H(t) − H(t − 1) exists (and is obviously zero in case G is finite) and h is called the asymptotic entropy of the random walk (X t ) t . Moreover, the Shannon-McMillan-Breiman type Theorem holds that 1 t |− log µ t (X t ) − H(t)| converges to zero almost surely. See [De, KV] for details.
In this note, we are interested in the case of a finite simple connected d-regular nonbipartite graph G and the quantitative growth property of H(t). Since G is connected and non-bipartite, one has µ t → π and H(t) ր H(π) = log |G|.
Here is an entropic characterization of cutoff phenomena for expander graphs. It is widely believed that every expander family (of transitive graphs) should exhibit cutoff.
It is well-known to experts that a cutoff phenomenon is generally related to a measure concentration phenomenon for − log µ t (X t ).
Corollary 4. Let G = {G} be an expander family of finite simple connected d-regular non-bipartite graphs. Assume that G has the following quantitative Shannon-McMillan-Breiman type property: For any δ, κ > 0 there is T 0 ∈ N such that every G ∈ G and t ≥ T 0 satisfy where N r (A) denotes the r-neighborhood of A ⊂ G. Then G exhibits cutoff.
Here is another auxiliary lemma about concavity of the square root.
Lemma 6. Let (Ω, A, P) be a probability space and B ⊂ A be a σ-subalgebra. For any random variable f ≥ 0 and a conditional expectation g = E f | B , one has Proof. By disintegration, the proof reduces to the case of f 0 ≥ 0 on a measure space (Y, η) and g 0 : Proof of Theorem 1. Recall from the beginning of the section that we want to estimate where .
The random variable f t satisfies f t ≥ 1 d and The difference 1 − E f t is caused by rare events where y ∈ supp µ t but x / ∈ supp µ t−1 , which only occur in case the random walks never backtrack. To relate the random variable f t to the functional analysis on ℓ 2 G, we consider the unit vectors ξ t := (µ t ) 1/2 in ℓ 2 G and observe that Hence, for any t ≤ T mix,2 (1 − ε), one has (38) E f 1/2 t ≤ P ξ t−1 ≤ ξ t−1 , π 1/2 + ρ < ρ + ε.
We will compare the value E f 1/2 t with that for the covering tree. Let's consider the covering of G by the d-regular tree T d and lift the random walk (X t ) t to the simple random walk (X t ) t on T d . We writef t :=μ t (X t )/μ t−1 (X t−1 ) and observe that Indeed, suppose X t−1 = x and X t = y. Then x and y are adjacent and P(X t−1 = x, X t = y) = µ t−1 (x)/d. There is a bijection q x,y from the lifts [x] of x onto the lifts [y] of y such that p and q x,y (p) are adjacent. Moreover, P(X t−1 = p,X t = q x,y (p)) =μ t−1 (p)/d. Thus at the event X t−1 = x and X t = y, one has By the concavity of the square root, one has Remark 7. In the above proof, we saw E f This implies that (f t ) t are asymptotically independent family and, by the central limit theorem, that Ramanujan graphs verify the quantitative Shannon-McMillan-Breiman type theorem (the statement without N δ log |G| in Corollary 4). In case of random walks on transitive graphs, one has E f 1/2 t = E g 1/2 t , where g t := P t (X t ,X 0 ) P t−1 (X t ,X 1 ) , and g t+1 ≈ E g t | X 1 , X t+1 . In view of Kaimanovich and Vershik's proof of the Shannon-McMillan-Breiman type theorem (Theorem 2.1 in [KV]), it would be interesting to know whether or not g t ≈ E g t | X 1 , . . . , X m for m = o(log |G|).