Conductance and Eigenvalue

We show the following. Theorem. Let M be an finite-state ergodic time-reversible Markov chain with transition matrix P and conductanceφ. Let λ ∈ (0, 1) be an eigenvalue of P. Then, φ + λ 6 1 This strengthens the well-known [4, 3, 2, 1, 5] inequality λ 6 1− φ2/2. We obtain our result by a slight variation in the proof method in [5, 4]; the same method was used earlier in [6] to obtain the same inequality for random walks on regular undirected graphs. AMarkov chain is a sequence of random variables {Xi}i>1 taking values in a finite set such that Pr[Xt = i | Xt−1 = j,Xt−2 = xt−2, · · · ,X0 = x0] = Pr[Xt = i | Xt−1 = j]. Let the state space of the Markov chain be [n] and let P = (Pij) be its n× n transition matrix: Pij = Pr[Xt = i | Xt−1 = j]. We will assume that the Markov chain is ergodic, that is, irreducible( for every pair of states i, j ∈ [n], P ij > 0 for some s) and aperiodic(for any state i ∈ [n], gcd{s : P ii > 0} = 1). Then, the Markov chain has a unique stationary distribution π: Pπ = π. We say that the Markov chain is time-reversible if it satisfies the following detailed balance condition: ∀i, j ∈ [n], Pijπj = Pjiπi (1) All Markov chains considered in this notewill be assumed to be finite-state ergodic and time-reversible. The conductance of a Markov chain with state space [n] is defined to be φ = min S⊂[n]: ∑ i∈S πi61/2 ∑ i∈S,j/ ∈S Pjiπi ∑ i∈S πi 1 The following theorem plays a central role in the theory of rapidly mixing Markov chains. Theorem ([5]). Let λ < 1 be an eigenvalue of the transition matrix of an ergodic time-reversible Markov chain with conductance φ. Then, λ 6 1− φ 2 2 . In this note we strengthen this inequality slightly. Theorem. Let λ ∈ (0, 1) be an eigenvalue of the transition matrix of an ergodic time-reversible Markov chain with conductance φ. Then, φ + λ 6 1 Such an inequality was derived by Radhakrishnan and Sudan [6] for the special case of random walks on regular undirected graphs. The purpose of this note is to show that their arguments (which were a slight variation on the arguments in [5, 4]) apply to finite-state ergodic time-reversible Markov chains as well. Proof. Let π be the stationary distribution of the chain with transition matrix P. Let f,g ∈ R. We will be thinking of f,g,π as vectors in R. Let 〈f,g〉 = ∑ i∈[n] fiπigi and ||f|| = √ 〈f, f〉. f is said to be proper if f 6= 0 and ∀i ∈ [n], fi > 0 and ∑ i∈[n]:fi>0 πi 6 1 2 We have the following two claims. Claim 1. For any proper f, φ||f|| 6 ||f|| − 〈f,P f〉 (2) Claim 2. For λ ∈ (0, 1), there exists a proper f such that 〈f,P f〉 > λ||f|| (3) Using (2) and (3), we obtain φ||f|| 6 ||f|| − λ||f|| from which the theorem follows. Proof of Claim 1. Permute the co-ordinates of f such that f1 > f2 > · · · > fr > 0 and fr+1 = · · · = fn = 0. (Note that ∑ i∈[r] πi 6 1/2.) We show that φ||f|| 6   ∑ i φ ∑ k∈[r] (fk − f 2 k+1) 