Improved small-set expansion from higher eigenvalues

Consider an irreducible reversible Markov chain on state space V , with |V | = n and invariant distribution π. Let 0 = λ1 ≤ λ2 ≤ · · ·λn ≤ 2 be the eigenvalues of its Laplacian operator. We give a simple spectral condition under which there exists a unit vector f ∈ L(V, π) with ‖f‖1 ≤ δ and 〈f, Lf〉 ≤ . (Using a standard Cheeger inequality, this implies the existence of a set S ⊆ V with measure at most O(δ) and expansion at most O( √ ).) As a consequence we show that for any k ∈ [n] and small α > 0, there is always a set S ⊆ V with measure at most O(k−1+α) and expansion at most √ λk logk n ·O(α−1/2). This essentially resolves a question of Arora, Barak, and Steurer, who obtained the same result with O(k−1/100) in place of O(k−1+α).