Geometric Analysis for the Metropolis Algorithm on Lipschitz Domains Persi Diaconis,∗ Gilles Lebeau,†‡ Laurent Michel§ Departments of Statistics and Mathematics, Stanford University and D´epartement de Math´ematiques, Universit´ede Nice Sophia-Antipolis Abstract This paper gives geometric tools: comparison, Nash and Sobolev inequalities for pieces of the relevent Markov operators, that give useful bounds on rates of convergence for the Metropolis algorithm. As an example, we treat the random placement of N hard discs in the unit square, the original application of the Metropolis algorithm. 1 Introduction and Results d Let Ω be a bounded, connected open subset of R . We assume that its boundary, ∂Ω, has Lipschitz d 1 R regularity. Let B1 be the unit ball of and ϕ(z) = 1 (z) so that ϕ(z)dz = 1. Let ρ(x) R vol(B1) B1 R be a measurable positive bounded function on Ω such that Ω ρ(x)dx = 1. For h ∈]0, 1], set x − y ρ(y) K (x, y) = h−dϕ min , 1 , (1.1) h,ρ h ρ(x) and let Th,ρ be the Metropolis operator associated with these data, that is, Z Th,ρ(u)(x) = mh,ρ(x)u(x) + Kh,ρ(x, y)u(y)dy, Ω Z (1.2) mh,ρ(x) = 1 − Kh,ρ(x, y)dy ≥ 0. Ω Then the Metropolis kernel Th,ρ(x, dy) = mh,ρ(x)δx=y + Kh,ρ(x, y)dy is a Markov kernel, the 2 operator Th,ρ is self-adjoint on L (Ω, ρ(x)dx), and thus the probability measure ρ(x)dx on Ω is ∗Supported in part by the National Science Foundation, DMS 0505673. †Supported in part by ANR equa-disp Blan07-3-188618 ‡Corresponding author: Parc Valrose 06108 Nice Cedex 02, France; [email protected] §Supported in part by ANR equa-disp Blan07-3-188618 1 n n stationary. For n ≥ 1, we denote by Th,ρ(x, dy) the kernel of the iterated operator (Th,ρ) . For n any x ∈ Ω, Th,ρ(x, dy) is a probability measure on Ω, and our main goal is to get some estimates n on the rate of convergence, when n → +∞, of the probability Th,ρ(x, dy) toward the stationary probability ρ(y)dy. A good example to keep in mind is the random placement of N non-overlapping discs of radius ε > 0 in the unit square. This was the original motivation for the work of Metropolis et al. [MRR+53]. One version of their algorithm goes as follows: from a feasable configuration, pick a disc (uniformly at random) and a point within distance h of the center of the chosen disc (uniformly at random). If recentering the chosen disc at the chosen point results in a feasable configuration, the change is made. Otherwise, the configuration is kept as it started. If N is fixed and ε and h are small, this gives a Markov chain with a uniform stationary distribution over all feasable configurations. The state space consists of the N centers corresponding to feasible configurations. It is a bounded domain with a Lipschitz boundary when N is small (see Section 4, Proposition 4.1). The scientific motivation for the study of random packing of hard discs as a way of understanding the apparent existence of a liquid/solid phase transition for arbitrarily large temperatures (for suitably large pressure) is clearly described in Uhlenbeck [Uhl68, Sect. 5, p. 18]. An overview of the large literature is in Lowen [L¨ow00].Entry to the zoo of modern algorithms to do the simulation (particularly in the dense case) with many examples is in Krauth [Kra06]. Further discussion, showing that the problem is still of current interest, is in Radin [Rad08]. We shall denote by g(h, ρ) the spectral gap of the Metropolis operator Th,ρ. It is defined as the largest constant such that the following inequality holds true for all u ∈ L2(ρ) = L2(Ω, ρ(x)dx). 2 2 1 kuk − hu, 1i ≤ hu − T u, ui 2 , (1.3) L2(ρ) L2(ρ) g(h, ρ) h,ρ L (ρ) or equivalently, Z Z 2 1 2 |u(x) − u(y)| ρ(x)ρ(y)dxdy ≤ Kh,ρ(x, y)|u(x) − u(y)| ρ(x)dxdy. (1.4) Ω×Ω g(h, ρ) Ω×Ω d Definition 1. We say that an open set Ω ⊂ R is Lipschitz if it is bounded and for all a ∈ ∂Ω d 0 there exists an orthonormal basis Ra of R , an open set V = V ×] − α, α[ and a Lipschitz map 0 η : V →] − α, α[ such that in the coordinates of Ra, we have 0 0 0 0 V ∩ Ω = y , yd < η(y ) , (y , yd) ∈ V ×] − α, α[ (1.5) V ∩ ∂Ω = y0, η(y0) , y0 ∈ V 0 . Our first result is the following: d Theorem 1.1. Let Ω be an open, connected, bounded, Lipschitz subset of R . Let 0 < m ≤ M < ∞ be given numbers. There exists h0 > 0, δ0 ∈]0, 1/2[ and constants Ci > 0 such that for any h ∈]0, h0], and any probability density ρ on Ω which satisfies for all x, m ≤ ρ(x) ≤ M, the following holds true. i) The spectrum of Th,ρ is a subset of [−1+δ0, 1], 1 is a simple eigenvalue of Th,ρ, and Spec(Th,ρ)∩ −2 [1 − δ0, 1] is discrete. Moreover, for any 0 ≤ λ ≤ δ0h , the number of eigenvalues of Th,ρ in 2 d/2 [1 − h λ, 1] (with multiplicity) is bounded by C1(1 + λ) . ii) The spectral gap g(h, ρ) satisfies 2 2 C2h ≤ g(h, ρ) ≤ C3h (1.6) 2 and the following estimates hold true for all integer n: n n −ng(h,ρ) (1 − g(h, ρ)) ≤ sup kTh,ρ(x, dy) − ρ(y)dykTV ≤ C4e . (1.7) x∈Ω The above results have to be understood as results for small h, other parameters of the problem being fixed. In particular, our estimates are certainly not sharp with respect to the dimension d of the space. For instance, a carefull look at the proof of estimate (1.7) shows that the constant C4 depends badly on the dimension d (if one tracks the dependance with respect to d, the bound d obtained by the Nash estimates can not be better than d , [DSC96]). Proving estimate on C4 with respect to the dimension would be of great interest. The next result will give some more information on the behavior of the spectral gap g(h, ρ) when h → 0. To state this result, let Z 1 Z 1 α = ϕ(z)z2dz = ϕ(z)|z|2dz = (1.8) d 1 d d + 2 and let us define ν(ρ) as the largest constant such that the following inequality holds true for all u in the Sobolev space H1(Ω): Z 2 2 1 αd 2 kukL2(ρ) − hu, 1iL2(ρ) ≤ |∇u| (x)ρ(x)dx, (1.9) ν(ρ) 2 Ω or equivalently, Z α Z |u(x) − u(y)|2ρ(x)ρ(y)dxdy ≤ d |∇u|2(x)ρ(x)dx. (1.10) Ω×Ω ν(ρ) Ω Observe that for a Lipschitz domain Ω, the constant ν(ρ) is well-defined thanks to Sobolev embed- ding. For a smooth density ρ, this number ν(ρ) > 0 is closely related to the unbounded operator 2 Lρ acting on on L (ρ). −αd ∇ρ Lρ(u) = (4u + .∇u) 2 ρ (1.11) 1 2 D(Lρ) = u ∈ H (Ω), −∆u ∈ L (Ω), ∂nu|∂Ω = 0 We now justify and explain the choice of domain in (1.11). Background for the following discussion and tools for working in Lipschitz domains is in [AF03]. When Ω has smooth boundary, standard elliptic regularity results show that for any u ∈ H1(Ω) 2 −→ such that −∆u ∈ L (Ω), the normal derivative of u at the boundary, ∂nu = n (x).∇u|∂Ω is well defined and belongs to the Sobolev space H−1/2(∂Ω). Here, we denote by −→n (x) the incoming unit normal vector to ∂Ω at a point x. In the case where ∂Ω has only Lipschitz regularity, the s Sobolev spaces H (∂Ω) are well defined for all s ∈ [−1, 1]. The trace operator, γ0(u) = u|∂Ω maps 1 1/2 1 H (Ω) onto H (∂Ω) = Ran(γ0), and its kernel is Ker(γ0) = H0 (Ω). Equipped with the norm 1/2 ∗ kukH1/2 = inf{kvkH1 , γ0(v) = u} it is an Hilbert space. Then, for any ϕ ∈ H (∂Ω) , there exists −1/2 R 1/2 a unique v ∈ H (∂Ω) such that ϕ(u) = ∂Ω vudσ for all u ∈ H (∂Ω) (where σ is the measure induced on the boundary). For v ∈ H−1/2(∂Ω), the support of v can be defined in a standard way. The trace operator acting on vector fields u ∈ (L2)d with div(u) ∈ L2, n 2 d 2 o −1/2 γ1 : u ∈ (L (Ω)) , div(u) ∈ L (Ω) → H (∂Ω), (1.12) 3 is then defined by the formula Z Z Z 1 div(u)(x)v(x)dx = − u(x).∇v(x)dx − γ1(u)v|∂Ωdσ(x), ∀v ∈ H (Ω). (1.13) Ω Ω ∂Ω 1 2 In particular, for u ∈ H (Ω) satisfying ∆u = div∇u ∈ L (Ω) we can define ∂nu|∂Ω = γ1(∇u) ∈ −1/2 1 H (∂Ω) and the set D(Lρ) is well defined. From (1.13) we deduce that for any u ∈ H (Ω) with ∆u ∈ L2 and any v ∈ H1(Ω) we have αd hL u, vi 2 = h∇u, ∇vi 2 + h∂ u, ρvi −1/2 1/2 .