JOURNALOFTHE AMERICANMATHEMATICALSOCIETY Volume23,Number3,July2010,Pages831–851 S0894-0347(10)00659-4 ArticleelectronicallypublishedonJanuary26,2010 PROOF OF ALDOUS’ SPECTRAL GAP CONJECTURE PIETROCAPUTO,THOMASM.LIGGETT,ANDTHOMASRICHTHAMMER 1. Introduction Spectral gap analysis plays an important role in the study of the convergence to equilibrium of reversible Markov chains. We begin by reviewing some well-known facts about Markov chains and their spectra. For more details we refer to [2]. 1.1. Finite state, continuous time Markov chains. Letusconsideracontinu- ous time Markov chain Z =(Zt)t(cid:2)0 with finite state space S and transition rates (q :i(cid:2)=j ∈S) such that q (cid:2)0. We will always assume that the Markov chain i,j i,j is irreducible and satisfies q =q for all i(cid:2)=j. i,j j,i Such a Markov chain is reversible with respect to the uniform distribution ν on S, whichistheuniquestationarydistributionofthechain. Theinfinitesimalgenerator L of the Markov chain is defined by (cid:2) Lg(i)= q (g(j)−g(i)), i,j j∈S where g : S → R and i∈ S. The matrix correspon(cid:3)ding to the linear operator L is thetransitionmatrixQ=(q ) ,whereq :=− q ,andthecorresponding i,j i,j i,i j(cid:3)=i i,j quadratic form is (cid:2) (cid:2) (cid:2) 1 g(i)Lg(i)= q g(i)(g(j)−g(i))=− q (g(j)−g(i))2. i,j 2 i,j i∈S i,j∈S i,j∈S Thus, −L is positive semi-definite and symmetric, which implies that its spectrum is of the form Spec(−L)={λ :0(cid:3)i(cid:3)|S|−1}, where i 0=λ0 <λ1 (cid:3) ··· (cid:3)λ|S|−1. The spectral gap λ is characterized as the largest constant λ such that 1 (cid:2) (cid:2) 1 (1.1) q (g(j)−g(i))2 (cid:2)λ g(i)2 2 i,j i,j∈S i∈S ReceivedbytheeditorsJune26,2009. 2010 MathematicsSubjectClassification. Primary60K35,60J27,05C50. Key words and phrases. Random walk, weighted graph, spectral gap, interchange process, symmetricexclusionprocess. ThefirstauthorwaspartiallysupportedbytheAdvancedResearchGrant“PTRELSS”ADG- 228032oftheEuropeanResearchCouncil. HethanksFilippoCesiforhelpfuldiscussions. PartialsupportfromNSFGrantDMS-0301795isacknowledged. (cid:2)c2010AmericanMathematicalSociety Reverts to public domain 28 years from publication 831 832 P.CAPUTO,T.M.LIGGETT,ANDT.RICHTHAMMER (cid:3) for all g : V → R with g(i) = 0. The significance of λ is its interpretation as i 1 the asymptotic rate of convergence to the stationary distribution: P (Z =j)=ν({j})+a e−λ1t+o(e−λ1t) for t→∞, i t i,j where typically a (cid:2)= 0 (and more precisely a > 0 for some i). For this reason i,j i,i 1 is sometimes referred to as the relaxation time of the Markov chain, and it is λ1 desirable to have an effective way of calculating λ . Aldous’ conjecture relates the 1 spectral gap of the random walk on a finite graph to that of more complicated processes on the same graph. This can be very important in applications, since generally speaking it is easier to compute or estimate (e.g. via isoperimetric in- equalities) the spectral gap of the random walk than that of the other processes considered, which have much larger state spaces. We say that the Markov chain with state space S and generator L is a sub- 2 2 process of the chain with state space S and generator L if there is a contraction 1 1 of S onto S , i.e. if there is a surjective map π :S →S such that 1 2 1 2 (1.2) L (f ◦π)=(L f)◦π for all f :S →R. 1 2 2 In this case, suppose that f is an eigenfunction of −L with eigenvalue λ. Then 2 −L (f◦π)=(−L f)◦π =λf◦πandf◦π (cid:2)=0forf (cid:2)=0,sof◦πisaneigenfunction 1 2 of −L with the same eigenvalue λ. Thus, 1 Spec(−L )⊂Spec(−L ), 2 1 and, in particular, the spectral gap of the first process is smaller than or equal to that of the second process. Identity (1.2) is an example of a so-called intertwining relation; see e.g. [6] for more details on such relations and their applications. 1.2. Random walk and interchange process on a weighted graph. LetG= (V,E)beanundirectedcompletegraphonnvertices; withoutlossofgeneralitywe assume that its vertex set is V = {1,...,n}. Furthermore G is a weighted graph in that we are given a collection of edge weights (or conductances) c (cid:2) 0, for xy xy = {x,y} ∈ E. Since we want the processes defined below to be irreducible, we will assume that the skeleton graph, i.e. the set of edges xy where c > 0, is xy connected. If we want to stress the dependence of one of the processes described below on the underlying weighted graph, we will write L(G) and λ (G) for its 1 generator and gap. Finally, we note that considering complete graphs only is no loss of generality, since edges with weight 0 can be thought of as being “absent”. 1.2.1. Random walk. The (1-particle) random walk on G is the Markov chain in which a single particle jumps from vertex x∈V to y (cid:2)=x at rate c ; see Figure 1. xy 1 5 2 4 3 Figure 1. Random walkonV ={1,2,3,4,5}. The pictureshows the underlying graph and a transition from state 1 to state 2. PROOF OF ALDOUS’ SPECTRAL GAP CONJECTURE 833 Formally, its statespace is SRW =V ={1,2,...,n}and itsgenerator is defined by (cid:2) LRWf(x)= c (f(y)−f(x)), for f :V →R, x∈V. xy y(cid:3)=x By Section 1.1, −LRW has n = |SRW| nonnegative eigenvalues and a positive spectral gap λRW >0. 1 1.2.2. Interchange process. In the interchange process, a state is an assignment of n labeled particles to the vertices of G in such a way that each vertex is occupied by exactly one particle. The transition from a state η to a state ηxy (occurring with rate c ) interchanges the particles at vertices x and y; see Figure 2. xy 1 3 5 5 2 1 5 1 3 4 2 4 2 4 3 Figure 2. Interchange process on V = {1,2,3,4,5}. The pic- ture shows the underlying graph and a transition from state η = (12345) to η1,2 = ητ = (12345). Note that in this 53142 1,2 53241 notation the first row refers to the labels. For aformal definition, letX denote the set of permutations of V ={1,...,n}, n and for η ∈ X and xy ∈ E let ηxy = ητ , where τ ∈ X is the transposition n xy xy n of x and y. The interchange process on G is the Markov chain with state space SIP =X and generator n (cid:2) LIPf(η)= c (f(ηxy)−f(η)), where f :SIP →R, η ∈SIP. xy xy∈E We use η to denote the label of the particle at x, while ξ =ξ (η) will be used to x i i denote the position of the particle labeled i. By Section 1.1, −LIP has |SIP| = n! nonnegativeeigenvaluesandapositivespectralgapλIP >0. Therandomwalkcan 1 be obtained as a sub-process of the interchange process by ignoring all particles apart from the one with label 1; more precisely the map π : SIP → SRW, π(η) := ξ (η) is a contraction in the sense of (1.2). Thus, 1 Spec(−LRW)⊂Spec(−LIP), and in particular, (1.3) λIP (cid:3)λRW . 1 1 1.3. Main result. Ourmainresultstatesthatinequality(1.3)is,infact,anequal- ity: Theorem 1.1. For all weighted graphs G, the interchange process and the random walk have the same spectral gap: (1.4) λIP(G)=λRW(G). 1 1 834 P.CAPUTO,T.M.LIGGETT,ANDT.RICHTHAMMER A weaker form of Theorem 1.1 involving only unweighted graphs had been con- jectured by Aldous around 1992, and since then it has been mostly referred to as Aldous’spectralgapconjectureintheliterature. Relatedobservationscanbefound in Diaconis and Shahshahani’s paper [9] and in the comparison theory developed by Diaconis and Saloff-Coste [8]. The problem has received a lot of attention in recent years—the conjecture was stated as an open problem on David Aldous’ web page [1] and in the influential monographs [2, 16]. In the meantime, various special cases have been obtained. The first class of graphs that was shown tosatisfy theconjecture is the class of un- weighted complete graphs (i.e. c =1 for all xy ∈E). Diaconis and Shahshahani xy computed all eigenvalues of the interchange process in this case using the irre- ducible representations of the symmetric group [9]. Similar results were obtained for unweighted star graphs in [12]. Recently, remarkable work of Cesi pushed this algebraic approach further to obtain the conjecture for all unweighted complete multipartite graphs [4]. Analternativeapproachbasedonrecursionwasproposedby HandjaniandJun- greis [13] (see also Koma and Nachtergaele [15] for similar results) who proved the conjecture for all weighted trees. The same ideas were recently used by Conomos and Starr [19] and Morris [18] to obtain an asymptotic version of the conjecture for boxes in the lattice Zd with unweighted edges. The basic recursive approach in [13] has been recently rephrased in purely algebraic terms; see [5, Lemma 3.1]. InordertoproveTheorem1.1,wedevelopageneralrecursiveapproachbasedon theideaofnetworkreduction; seeSection2. Themethod,inspiredbythetheoryof resistive networks, allows us to reduce the proof of the theorem to the proof of an interesting comparison inequality for random transposition operators on different weighted graphs; see Theorem 2.3. After a preliminary version [3] of this paper appeared, we learned that the same recursive strategy had been discovered around the same time independently, and from a slightly different perspective, by Dieker [10]. The comparison inequality alluded to above was conjectured to hold in both [3] and [10]. The comparison inequality will be proved in Section 3. The main idea for this proof is a decomposition of the associated matrix into a covariance matrix and a correction matrix (a Schur complement). A delicate analysis based on block decompositions corresponding to suitable cosets of the permutation group reveals that the correction matrix is nonnegative definite. Some immediate consequences of Theorem 1.1 for other natural Markov chains associated to finite weighted graphs are discussed in Section 4. We end this introductory section with a collection of known properties of the spectrum of the interchange process that can be deduced from the algebraic ap- proach. We refer to [9, 12, 4] and references therein for more details. These facts are not neededin what follows and thereader may safely jump tothe next section. However, we feel that the algebraic point of view provides a natural decomposition of the spectrum that is worth mentioning. 1.4. Structure of the spectrum of −LIP. In Section 1.2.2 we saw that Spec(−LRW)⊂Spec(−LIP). PROOF OF ALDOUS’ SPECTRAL GAP CONJECTURE 835 One can go a little further and show that if 0 = λRW < λRW (cid:3) ··· (cid:3) λRW are 0 1 n−1 the eigenvalues of −LRW, then for k (cid:2)0 and 1(cid:3)i <···<i (cid:3)n−1, 1 k (1.5) λRW +···+λRW ∈Spec(−LIP). i1 ik Thecorrespondingeigenfunctionistheantisymmetricproductofthek one-particle eigenfunctions of λRW,...,λRW. In particular, the eigenvalue i1 ik (cid:2) (1.6) λRW +···+λRW =Tr(−LRW)=2 c 1 n−1 xy xy∈E is associated withfunctions that are antisymmetric in all particles, i.e. multiples of the alternating function h(η) = sign(η). (This also follows directly from h(ηxy)− h(η) = −2h(η).) From the representation theory of the symmetric group one can compute (see below) the multiplicity of all eigenvalues of the form (1.5), and one findsthattheoverwhelmingmajority(forlargen)ofthespectrumof−LIP arenot of this form. (cid:4) The vector space of functions f :X →R is equivalent to a direct sum H , n α α where α ranges over all (equivalence classes of the) irreducible representations of the symmetric group. Since the latter are in one-to-one correspondence with the partitions of n, one can identify α with a Young diagram α = (α1,α2,.(cid:3)..), where the α form a nonincreasing sequence of nonnegative integers such that α =n. i (cid:4) i i Each subspace H is in turn a direct sum H = dα Hj, of subspaces Hj, α α j=1 α α each of dimension dα, where the positive integer dα is the dim(cid:3)ension of the irre- duciblerepresentationα. Inparticular, thenumbersd satisfy (d )2 =n!. The α α α subspaces Hj are invariant for the actionof the generator −LIP, so that −LIP can α be diagonalized within each Hj. Subspace Hi will produce d eigenvalues λ (α), α α α k k =1,...,d . Some of these may coincide if the weights have suitable symmetries α (for instance, if G is the complete graph with c = 1 for all xy ∈ E, then they xy all coincide and −LIP is a multiple of the identity matrix in each H ; cf. [9]) but α in the general weighted case they will be distinct. On the other hand, for a given α, the eigenvalues coming from Hi are identical to those coming from Hj, for all α α i,j = 1,...,d , so that each eigenvalue λ (α) will appear with multiplicity d in α k α the spectrum of −LIP. Moreover, using known expressions for the characters of transpositions, one can compute explicitly the sum (cid:2)dα λ (α), k k=1 for every irreducible representation α, as a function of the edge weights. For in- stance, when α is the partition (n−1,1,0,...), which has d =n−1, one obtains α the relation (1.6). The trivial partition (n,0,...) has dimension 1 and the only eigenvalue is 0. This is the space of constant functions. Similarly, the alternating partition((cid:3)1n,0...)(nonesandthenallzeros),hasdimension1andtheonlyeigen- value is 2 c . It can be shown that the eigenvalues of the form (1.5) come xy∈E xy (cid:5) (cid:6) fromtheL-shapedpartitionsα=(n−k,1k,0,...),eachwithdimensiond = n−1 . (cid:3) (cid:5) (cid:6) (cid:5)α (cid:6)k So the total number of eigenvalues of the form (1.5) is n−1 n−1 2 = 2(n−1) . k=0 k n−1 Finally, usingknownrelationshipsbetweenconjugateirreduciblerepresentations (see e.g. [14, 2.1.8], [5, (2.12)]), one can show that the spectrum of −LIP can be 836 P.CAPUTO,T.M.LIGGETT,ANDT.RICHTHAMMER decomposed into pairs of eigenvalues λ,λ(cid:4) such that (cid:2) (cid:4) λ+λ =2 c , xy xy∈E where λ,λ(cid:4) are associated with conjugate Young diagrams. 2. A recursive approach based on network reduction Given a weighted graph G = (V,E) as above and a point x ∈ V, we consider the reduced network obtained by removing the vertex x. This gives a new graph G with vertex set V := V \{x}, edge set E = {yz ∈ E : y,z (cid:2)= x}, and edge x x x conductances (cid:7)c (cid:2)c defined by yz yz c c (2.1) (cid:7)c =c +c∗,x, c∗,x := (cid:3) xy xz , yz yz yz yz c w∈Vx xw for yz ∈ E . We refer to G as the reduction of G at x or simply as the reduced x x graph at x. This is the general version of more familiar network reductions such as series resistance (from 3 to 2 vertices) or star-triangle transformations (from 4 to 3 vertices); see Figure 3. We refer to [11, 17, 2] for the classical probabilistic point of view on electric networks. 5 4 4 1 1 =⇒ 3 3 2 2 Figure 3. Reduction of a 5-vertex graph to a 4-vertex graph at x=5. 2.1. Random walk on the reduced network. We first show that the spectral gap of the random walk on the reduced network is not smaller than the original random walk spectral gap: Proposition 2.1. The spectral gaps of the random walks on a weighted graph G and the corresponding reduced graph G satisfy x λRW(G )(cid:2)λRW(G). 1 x 1 Proof. We will use the shorthand notation L = LRW(G) and L = LRW(G ) for x x the generators of the two random walks. We first note that, for any graph G, λRW(G) can be characterized as the largest constant λ such that 1 (cid:2) (cid:2) (2.2) (Lg(z))2 (cid:2) −λ g(z)Lg(z) z∈V z∈V PROOF OF ALDOUS’ SPECTRAL GAP CONJECTURE 837 h(cid:3)olds for all g : V (cid:3)→ R. To see this, observe that, for any g,h : V → R, h(z)Lg(z) = g(z)Lh(z). Thus, taking h = Lg, the left hand side z∈V z∈V of (2.2) coincides with the quadratic form (cid:2) g(z)L2g(z), z∈V and (2.2) says that L2+λL is nonnegative definite. Taking a basis which makes L diagonal, one sees that this holds iff λ(cid:3)λRW(G). 1 To prove the proposition, take a function g : V → R harmonic at x, i.e. such that Lg(x)=0. Then (cid:3) c g(y) (2.3) g(x)= (cid:3)y∈Vx xy . c w∈Vx xw For any z ∈V , from (2.3) we have x (cid:2) Lg(z)= c [g(y)−g(z)]+c [g(x)−g(z)] zy zx y(cid:2)∈Vx(cid:5) (cid:6) = c +c∗,x [g(y)−g(z)]. zy zy y∈Vx In other words (cid:8) L g(z), z ∈V , Lg(z)= x x 0, z =x. Applying (2.2) to this function, we have (cid:2) (cid:2) (cid:2) (L g(z))2 = (Lg(z))2 (cid:2) −λRW(G) g(z)Lg(z) x 1 z∈Vx z∈V (cid:2) z∈V =−λRW(G) g(z)L g(z). 1 x z∈Vx Since the function g is arbitrary on V , using (2.2) again, this time for the graph x G , we obtain λRW(G )(cid:2)λRW(G). (cid:4) x 1 x 1 Proposition 2.1 generalizes the observation in [13] that if G is a graph with a vertex x of degree 1 (i.e. only one edge out of x has positive weight), then the spectral gapof therandom walkcannot decreasewhenwe cancelx andremove the only edge connecting it to the rest of G. (In that case (cid:7)c =c since x has degree yz yz 1.) We end this subsection with a side remark on further relations between the generators L = LRW(G) and L = LRW(G ). When we remove a vertex, it is x x interesting to compare the energy corresponding to the removed branches with the energy coming from the new conductances. The following identity can be obtained with a straightforward computation. Lemma 2.2. For any fixed x∈V and any g :V →R, (cid:2) (cid:2) 1 c [g(y)−g(x)]2 = c∗,x[g(y)−g(z)]2+ (cid:3) (Lg(x))2. xy yz c y∈Vx yz∈Ex y(cid:3)=x xy 838 P.CAPUTO,T.M.LIGGETT,ANDT.RICHTHAMMER ConsidertheoperatorL(cid:7) definedbyL(cid:7) g(x)=0andL(cid:7) g(z)=L g(z)forz (cid:2)=x, x x x x where g : V → R. Then L(cid:7) is the generator of the random walk on G ∪{x}, x x where x is an isolated vertex. Lemma 2.2 implies that the quadratic form of −L(cid:7) x is dominated by the quadratic form of −L. It follows from the Courant-Fisher min-max theorem that if λ(cid:7)0 (cid:3) ··· (cid:3) λ(cid:7)n−1 denote the eigenvalues of −L(cid:7)x, then λ(cid:7) (cid:3)λRW(G), i=0,...,n−1. Note that this is not in contradiction to the result i i (cid:7) (cid:7) in Proposition 2.1 since, due to the isolated vertex x, one has λ = λ = 0 and 0 1 λ(cid:7) = λRW(G ), k = 1,...,n−2. While the bound in Proposition 2.1 will be k+1 k x sufficient forour purposes, it isworthpointingoutthat, asobservedin[10], atthis point standard results on interlacings can be used to prove the stronger statement λRW(G)(cid:3)λRW(G )(cid:3)λRW(G), j =1,...,n−2. j j x j+1 2.2. Octopus inequality. The following theorem summarizes the main technical ingredient we shall need. Here ν is the uniform(cid:9) probability measure on all per- mutations X , and we use the notation ν[f] = fdν. The gradient ∇ is defined n by ∇ f(η)=f(ηxy)−f(η). xy Theorem 2.3. For any weighted graph G on |V| = n vertices, for every x ∈ V and f :X →R, n (cid:2) (cid:2) (2.4) c ν[(∇ f)2] (cid:2) c∗,xν[(∇ f)2]. xy xy yz yz y∈Vx yz∈Ex Note that if f(η) = g(ξ ) is a function of one particle, then a simple computation 1 gives 2 ν[(∇ f)2]= (g(u)−g(v))2, uv ∈E, uv n so that this special case of Theorem 2.3 is contained in Lemma 2.2. The identity in Lemma 2.2 also shows that in this case the inequality is saturated by functions thatareharmonicatx. Ontheotherhand, thegeneralcaserepresentsanontrivial comparison inequality between a weighted star graph and its complement, with weights defined by (2.1). Inspired by its tentacular nature, we refer to the bound (2.4) as the octopus inequality. We will give a proof of Theorem 2.3 in Section 3. 2.3. Reformulation of the conjecture. We shall use the following convenient notation: AsaboveletνdenotetheuniformprobabilitymeasureXn,∇(cid:3)thegradient, and b a generic edge, whose weight is denoted c . In this way LIP = c ∇ and b b b b the Dirichlet form −ν[fLIPf] is (cid:2) 1 E(f)= c ν[(∇ f)2]. 2 b b b The spectral gap λIP is the best constant λ so that for all f :X →R, 1 n (2.5) E(f)(cid:2)λVar (f), ν where Var (f) = ν[f2]−ν[f]2 is the variance of f w.r.t. ν. In order to get some ν hold on the eigenvalues of the interchange process that are not eigenvalues of the random walk, we introduce the vector space H={f :X →R:ν[f|ξ ]=0 for all i∈V} n i ={f :X →R:ν[f|η ]=0 for all x∈V}, n x PROOF OF ALDOUS’ SPECTRAL GAP CONJECTURE 839 where ν[·|ξ ] and ν[·|η ] are the conditional expectationsgiven the position of the i x particlelabelediandgiventhelabeloftheparticleatx,respectively. Theequality in the definition of H is a consequence of ν[·|ξ ](η)=ν[·|ξ =x]=ν[·|η =i]=ν[·|η ](η), i i x x where η ∈X is such that ξ (η)=x. Note that for every i, n i (cid:2) ν[LIPf|ξ =x]= c (ν[f|ξ =y]−ν[f|ξ =x]) , i xy i i y(cid:3)=x for all f : X → R and x ∈ V. In particular, H is an invariant subspace for n −LIP, and if f ∈/ H is an eigenfunction of −LIP with eigenvalue λ, then ν[f|ξ ](cid:2)= i 0 for some i, and ν[f|ξ = x], x ∈ V, is an eigenfunction of −LRW with the i same eigenvalue λ. It follows that H contains all eigenfunctions corresponding to eigenvaluesinSpec(−LIP)\Spec(−LRW). Therefore,ifμIP(G)denotesthesmallest 1 eigenvalue of −LIP associated to functions in H (i.e. the best constant λ in (2.5) restricting to functions f ∈H), then for every graph G one has λIP(G)=min{λRW(G),μIP(G)}. 1 1 1 The assertion λIP(G)=λRW(G) of Theorem 1.1 becomes then equivalent to 1 1 (2.6) μIP(G)(cid:2)λRW(G). 1 1 In the rest of this section we show how the network reduction idea, assuming the validity of Theorem 2.3, yields a proof of Theorem 1.1. 2.4. Proof of Theorem 1.1. We use the notation from the previous sections. In particularwewriteλRW(G )andλIP(G )forthespectralgapsoftherandomwalk 1 x 1 x and the interchange process in the network reduced at x. Let us first show that Theorem 2.3 implies an estimate of μIP(G). 1 Proposition 2.4. For an arbitrary weighted graph G (2.7) μIP(G)(cid:2) maxλIP(G ). 1 x∈V 1 x Proof. Let f ∈H and x∈V. Since ν[f|η ]=0, we have x ν[f2]=Var (f)=ν[Var (f|η )], ν ν x where Var (f|η ) is the variance w.r.t. ν[·|η ]. For a fixed value of η , ν[·|η ] is ν x x x x the uniform measure on the permutations on V = V \{x}. Therefore using the x spectral gap bound (2.5) on the graph G , we have x (cid:2) λIP(G )Var (f|η )(cid:3) 1 (c +c∗,x)ν[(∇ f)2|η ], 1 x ν x 2 b b b x b:b(cid:3)(cid:6)x with c∗,x defined by (2.1). Taking the ν-expectation, we obtain b (cid:2) λIP(G )ν[f2](cid:3) 1 (c +c∗,x)ν[(∇ f)2]. 1 x 2 b b b b:b(cid:3)(cid:6)x From Theorem 2.3, (cid:2) (cid:2) c∗,xν[(∇ f)2](cid:3) c ν[(∇ f)2]. b b b b b:b(cid:3)(cid:6)x b:b(cid:6)x Therefore, (2.8) λIP(G )ν[f2](cid:3)E(f). 1 x 840 P.CAPUTO,T.M.LIGGETT,ANDT.RICHTHAMMER Since x ∈ V and f ∈ H were arbitrary, this proves that, for every x ∈ V, μIP(G)(cid:2)λIP(G ), establishing the inequality (2.7). (cid:4) 1 1 x Propositions 2.1 and 2.4 allow us to conclude the proof by recursion. Indeed, note that λIP(G) = λRW(G) is trivially true when G = b is a single weighted 1 1 edge b. (When n = 2, the random walk and the interchange process are the same 2-state Markov chain.) If G is a weighted graph on n vertices, we assume that λIP(G(cid:4)) = λRW(G(cid:4)) holds on every weighted graph G(cid:4) with n − 1 vertices, in 1 1 particular on G . Therefore x μIP(G)(cid:2) maxλIP(G )=maxλRW(G )(cid:2)λRW(G), 1 x∈V 1 x x∈V 1 x 1 wherewealsohaveusedPropositions2.1and2.4. Thuswehaveshown(2.6),which is equivalent to λIP(G)=λRW(G). 1 1 3. Proof of the octopus inequality For the proof of Theorem 2.3 we slightly change our notation as follows. We set V = {0,1,...,n−1} and x = 0. The only rates appearing in (2.4) are c , so we 0i set (cid:2) c :=c for 1(cid:3)i, c :=− c , i 0i 0 i (cid:2) i(cid:2)(cid:2)1 and c:= c2+ c c . i i j 1(cid:3)i(cid:3)n−1 1(cid:3)i<j(cid:3)n−1 Note that c <0 and 0 (cid:2) (cid:2) (3.1) c =0, c=− c c , and c∗,0 =−cicj. i i j ij c i(cid:2)0 0(cid:3)i<j 0 Using this shorthand notation, the octopus inequality (2.4) simplifies to (cid:2) (cid:2) (3.2) − c c (f(ητ )−f(η))2 (cid:2)0, i j ij 0(cid:3)i<j η where τ denotes the transposition of i,j ∈ V, i.e. ητ = ηij. Thus it suffices to ij ij show that the matrix C defined by ⎧ ⎨ c if η =η(cid:4), (3.3) Cη,η(cid:2) =⎩ cicj if ητij =η(cid:4), 0 otherwise is positive semi-definite for every n and all rates c1,...,cn−1 (cid:2)0. 3.1. Decomposition of the matrix C. In the following we write A (cid:2) B if the sameinequalityholdsforthecorrespondingquadraticforms,i.e.ifA−B ispositive semi-definite. Obviously, this defines apartial order and we will repeatedlyuse the following simple facts for square matrices A,B and a real number a: A(cid:2)0, B (cid:2)0 ⇒ A+B (cid:2)0; A(cid:2)0, a(cid:2)0 ⇒aA(cid:2)0; (cid:13) (cid:14) A 0 (cid:2)0 ⇔ A,B (cid:2)0. 0 B