Improved Strong Spatial Mixing for Colorings on Trees
Charilaos Efthymiou, Andreas Galanis, Thomas P. Hayes, Daniel Stefankovic, Eric Vigoda
aa r X i v : . [ c s . D M ] S e p Improved Strong Spatial Mixing for Colorings on Trees
Charilaos Efthymiou ∗ Andreas Galanis † Thomas P. Hayes ‡ Daniel ˇStefankoviˇc § Eric Vigoda ¶ September 17, 2019
Abstract
Strong spatial mixing (SSM) is a form of correlation decay that has played an essentialrole in the design of approximate counting algorithms for spin systems. A notable exampleis the algorithm of Weitz (2006) for the hard-core model on weighted independent sets. Westudy SSM for the q -colorings problem on the infinite ( d +1)-regular tree. Weak spatialmixing (WSM) captures whether the influence of the leaves on the root vanishes as theheight of the tree grows. Jonasson (2002) established WSM when q > d + 1. In contrast, inSSM, we first fix a coloring on a subset of internal vertices, and we again ask if the influenceof the leaves on the root is vanishing. It was known that SSM holds on the ( d + 1)-regulartree when q > αd where α ≈ . ... is a constant that has arisen in a variety of resultsconcerning random colorings. Here we improve on this bound by showing SSM for q > . d .Our proof establishes an L contraction for the BP operator. For the contraction we boundthe norm of the BP Jacobian by exploiting combinatorial properties of the coloring of thetree. Consider random q -colorings of the complete tree T h of height h with branching factor d . Doesthe influence of the leaves on the root decay to zero in the limit as the height grows? If so, thiscorresponds to weak spatial mixing, which we will define more precisely momentarily.Now suppose we fix the coloring τ for a subset of internal vertices. Is it still the case thatthe influence of the leaves on the root decay to zero as the height grows? One might intuitivelyexpect that these internal “agreements” defined by τ only help in the sense that the influenceof the leaves decrease, however this problem is much more challenging; it corresponds to strongspatial mixing, which is the focus of this paper.For statistical physics models, the key algorithmic problems are the counting problem ofestimating the partition function and the problem of sampling from the Gibbs distribution,which corresponds to the equilibrium state of the system. Strong spatial mixing ( SSM ) is a keyproperty of the system for the design of efficient counting/sampling algorithms. ∗ Department of Computer Science, University of Warwick, UK. Supported by the Centre of Discrete Mathe-matics and its Applications (DIMAP), University of Warwick, EPSRC award EP/D063191/1. † Department of Computer Science, University of Oxford, UK. The research leading to these results has re-ceived funding from the European Research Council under the European Union’s Seventh Framework Programme(FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors’ views and not the viewsof the ERC or the European Commission. The European Union is not liable for any use that may be made ofthe information contained therein. ‡ Department of Computer Science, University of New Mexico, USA. Partially supported by NSF CAREERaward CCF-1150281. § Department of Computer Science, University of Rochester, USA. Research supported in part by NSF grantCCF-1563757. ¶ School of Computer Science, Georgia Institute of Technology, USA. Research supported in part by NSFgrants CCF-1617306 and CCF-1563838. SM has a variety of algorithmic implications. A direct consequence of SSM on amenablegraphs, such as the integer lattice Z d , is fast mixing of the Glauber dynamics, which is thesimple Markov chain that updates the spin at a randomly chosen vertex in each step, see,e.g. [21, 22, 6, 9, 14, 4, 3]. SSM also plays a critical role in the efficiency of correlation-decaytechniques of Weitz [26] which yields an
FPTAS for the partition function of the hard-coremodel in the tree uniqueness region; this approach has been extended to 2-spin antiferromagneticmodels [18] and other interesting examples, e.g., [19]; note, the approach of Barvinok [1] utilizinga zero-free region of the partition function in the complex plane has recently been extended tothe same range of parameters for the hard-core model [23, 25].The fundamental question in statistical physics is the uniqueness/non-uniqueness phasetransition which corresponds to whether long-range correlations persist or die off, in the limitas the volume of the system tends to infinity. In the uniqueness region the correlations die off,which corresponds to weak spatial mixing ( WSM ). While
WSM (or equivalently uniqueness) isa notoriously challenging problem on the 2-dimensional integer lattice Z (e.g., see the recentbreakthrough work of Beffara and Duminil-Copin [2] for the ferromagnetic Potts model), thecorresponding WSM problem on the infinite ( d + 1)-regular tree T d , known as the Bethe lattice,is typically simpler since it can be analyzed using recursions due to the absence of cycles (e.g.,see Kelly [17] for the hard-core model). However, for the colorings problem, which is the focusof this paper, even WSM is far from trivial on the regular tree [16]. In fact, for the closelyrelated antiferromagnetic Potts model the precise range of parameters for
WSM is only knownfor fixed values of q, d [11].The focus of this paper is on these correlation decay properties on the infinite ( d + 1)-regulartree T d for the colorings problem. We give an informal definition of WSM and
SSM , and referthe interested reader to Section 2 for formal definitions.Let T h denote the complete tree of height h where all internal vertices have degree d + 1.For integer q ≥
3, let µ h denote the uniform distribution over proper (vertex) q -colorings of T h . Consider a pair of sequences of colorings ( η h ) and ( η ′ h ) for the leaves of T h . Let p h and p ′ h denote the marginal probability that the root receives a specific color c under µ h conditional onthe leaves having the fixed coloring η h and η ′ h , respectively. Roughly, if lim h →∞ | p h − p ′ h | = 0 forall sequences ( η h ) , ( η ′ h ) and colors c , then we say WSM holds (see also Section 2). Jonasson [16]proved that
WSM holds when q ≥ d + 2. When q ≤ d + 1, the pair of boundary conditions canactually “freeze” the color at the root; moreover, Brightwell and Winkler [5] showed that thereare multiple semi-translation invariant Gibbs measures on T d when q ≤ d .Now consider an arbitrary coloring τ for a subset S ⊂ T d . Let r h and r ′ h denote the marginalprobability that the root receives color c under µ h conditional on η h ∪ τ and η ′ h ∪ τ , respectively.If these limits are the same then we say SSM holds. The challenge of establishing
SSM isillustrated by the fact that if
WSM holds then we know that lim h →∞ p h = 1 /q but that is notnecessarily the case in the SSM setting.Ge and ˇStefankoviˇc [13] proved that
SSM holds on T d when q > αd where α ≈ . ... isthe root of α exp(1 /α ) = 1. Gamarnik, Katz, and Misra [12] extended this result to arbitrarytriangle-free graphs of maximum degree d , under the same condition on q . Recent work of Liu,Sinclair, and Srivistava [20] builds upon [12] together with the approximate counting approachof [1, 23] to obtain an FPTAS for counting colorings of triangle-free graphs when q > αd .Prior to these works, Goldberg, Martin, and Paterson [14] established the above form of
SSM on triangle-free amenable graphs, also when q > αd . In addition to the above results, thethreshold α ≈ . . . . has arisen in numerous rapid mixing results, e.g., [7, 15, 8].Our main result presents the first substantial improvement on the 1 . ... threshold of [13],we establish SSM on the tree when q > . d . We state a somewhat informal version of ourmain theorem here, the formal version will be given once we define more precisely SSM , cf.Theorem 3 below. Roughly, a graph is amenable if for every subset S of vertices, the neighborhood satisfies | N ( S ) | ≤ poly( | S | ) heorem 1 (Informal version of Theorem 3) . There exists an absolute constant β > suchthat, for all positive integers q, d satisfying q ≥ . d + β , the q -coloring model exhibits strongspatial mixing on the regular tree T d . We remark that the constant 1 .
59 in Theorem 1 can be replaced with any α ′ > α ′ exp (cid:16) α ′ (cid:17) exp (cid:16) − α ′ − (cid:0) α ′ − (cid:1) (cid:17) < , the smallest such value up to four decimal digits is 1 . SSM inSection 2 and stating the formal version of Theorem 1. We then present detailed proofs of thethree main lemmas in Section 3.
Let q ≥ G = ( V, E ) be a graph. A proper q -coloring of G is an assignment σ : V → [ q ] such that for every ( u, v ) ∈ E it holds that σ ( u ) = σ ( v ). We use Ω G to denote theset of all proper q -colorings of G and µ G to denote the uniform probability distribution on Ω G (provided that Ω G is non-empty).For σ ∈ Ω G and a set Λ ⊂ V , we use σ Λ to denote the restriction of σ to Λ. When Λ consistsof a single vertex v , we will often use the shorthand σ v to denote the color of v under σ . We saythat an assignment η : Λ → [ q ] is extendible if there exists a coloring σ ∈ Ω G such that σ Λ = η .We can now formally define SSM . Definition 2.
Let ζ : Z ≥ → [0 , be a real-valued function on the positive integers.The q -coloring model exhibits strong spatial mixing , denoted SSM , on a finite graph G =( V, E ) with decay rate ζ ( · ) iff for every v ∈ V , for every Λ ⊂ V , for any two extendibleassignments η, η ′ : Λ → [ q ] and any color c ∈ [ q ] it holds that (cid:12)(cid:12) µ G ( σ v = c | σ Λ = η ) − µ G ( σ v = c | σ Λ = η ′ ) (cid:12)(cid:12) ≤ ζ (cid:0) dist( v, ∆) (cid:1) , (1) where ∆ ⊆ Λ denotes the set of vertices where η and η ′ disagree.In the case where G is infinite, we say that the q -coloring model exhibits strong spatial mixingon G with decay rate ζ ( · ) if it exhibits strong spatial mixing on every finite subgraph of G withdecay rate ζ ( · ) . The definition of weak spatial mixing has one modification: in the RHS of (1) we replacedist( v, ∆) by the weaker condition dist( v, Λ).
WSM says that the influence of a pair of boundaryconditions decays at rate ζ ( · ) in the distance to the boundary Λ. In SSM the pair of boundaries η, η ′ might only differ on a subset ∆ ⊂ Λ; do these fixed “agreements” on Λ \ ∆ influence themarginal at v ? If SSM holds then the difference in the marginal at v decays at rate ζ ( · ) in thedistance to the “disagreements” in η, η ′ .With these definitions in place, we are now ready to give the formal version of Theorem 1. Theorem 3.
There exists an absolute constant β > such that, for all positive integers q, d satisfying q ≥ . d + β , the q -coloring model exhibits strong spatial mixing on the regular tree T d with exponentially decaying rate.That is, there exist constants α, C > and a function ζ satisfying ζ ( ℓ ) ≤ C exp( − αℓ ) forall integers ℓ ≥ such that for all finite subtrees T of T d the q -coloring model exhibits strongspatial mixing on T with decay rate ζ . Proof Approach
For a set Λ ⊂ V and an extendible assignment η : Λ → [ q ], we use π G,v,η to denote the q -dimensional probability vector whose entries give the marginal distribution of colors at v under the boundary condition η , i.e., for a color c ∈ [ q ], the c -th entry of π G,v,η is given by µ G ( σ v = c | σ Λ = η ).The key ingredient to prove Theorem 3 is the following. Theorem 4.
There exist absolute constants β > and U ∈ (0 , such that the following holdsfor all positive integers q, d satisfying q ≥ . d + β .Let T = ˆ T d,h,ρ be the d -ary tree with height h rooted at ρ , Λ be a subset of the vertices of T , and η, η ′ : Λ → [ q ] be two extendible assignments of T with dist( ρ, ∆) ≥ where ∆ ⊆ Λ isthe set of vertices where η and η ′ disagree. Let v , . . . , v d be the children of ρ and for i ∈ [ d ] let T i = ( V i , E i ) be the subtree of T rooted at v i which consists of all descendants of v i in T . Then (cid:13)(cid:13) π − π ′ (cid:13)(cid:13) ≤ U max i ∈ [ d ] (cid:13)(cid:13) π i − π i ′ (cid:13)(cid:13) , where π = π T,ρ,η , π ′ = π T,ρ,η ′ and for i ∈ [ d ] we denote π i = π T i ,v i ,η (Λ ∩ V i ) , π ′ i = π T i ,v i ,η ′ (Λ ∩ V i ) . Intuitively, Theorem 4 says that disagreements between η and η ′ have smaller impact onthe marginals as we move upwards on the tree. More precisely, the marginals of the root under η and under η ′ are closer in L distance than the distance between the marginals of any child(under the induced distributions on the subtrees hanging from them).Using Theorem 4, the proof of Theorem 3 of strong spatial mixing follows from ratherstandard considerations, the proof can be found in Section 7. In the following section, we focuson the more interesting proof of Theorem 4 and explain the new aspects of our analysis. In this section, we lay down the main technical steps in proving Theorem 4. In particular, wewill assume throughout that, for appropriate integers q, d, h , T = ˆ T d,h,ρ is the d -ary tree withheight h rooted at ρ , Λ is a subset of the vertices of T , and η, η ′ : Λ → [ q ] are two extendibleassignments of T with dist( ρ, ∆) ≥ ⊆ Λ is the set of vertices where η and η ′ disagree.We als let v , . . . , v d be the children of ρ and for i ∈ [ d ] let T i = ( V i , E i ) be the subtree of T rooted at v i which consists of all descendants of v i in T .To prove Theorem 4, we will use tree recursions to express the marginal at the root in termsof the marginals at the children (as in previous works on WSM/SSM, see, e.g., [5, 13, 11]). Thisrecursion is the well-known Belief Propagation (BP) equation [24]; our proof of Theorem 4 willbe based on bounding appropriately the gradient of the BP equations. The new ingredient inour analysis is that we incorporate the combinatorial structure of agreements close to the rootinto a refined L analysis of the gradient.Prior to delving into the analysis, we first describe the BP equation for the colorings model.Following the notation of Theorem 4, let π = π T,ρ,η , π ′ = π T,ρ,η ′ be the marginal distributionsat the root of the tree T under the boundary conditions η and η ′ , respectively. Similarly, for i ∈ [ d ], let π i , π ′ i be the marginals at the root v i of the subtree T i under η (Λ ∩ V i ) and η ′ (Λ ∩ V i ),respectively.We can now relate the distribution π with the distributions { π i } i ∈ [ d ] (and similarly, π ′ withthe distributions { π ′ i } i ∈ [ d ] ) as follows. For q -dimensional probability vectors x , . . . , x d and acolor c ∈ [ q ], let f c be the function f c ( x , . . . , x d ) = Q i ∈ [ d ] (cid:0) − x i,c (cid:1)P j ∈ [ q ] Q i ∈ [ d ] (cid:0) − x i,j (cid:1) , (2)4here, for i ∈ [ d ] and j ∈ [ q ], x i,j denotes the j -th entry of the vector x i . Then, with π c and π ′ c denoting the c -th entries of π and π ′ , we have that π c = µ T ( σ ρ = c | σ Λ = η ) = f c ( π , . . . , π d ) ,π ′ c = µ T ( σ ρ = c | σ Λ = η ′ ) = f c ( π ′ , . . . , π ′ d ) . (3)The functions { f c } c ∈ [ q ] correspond to the BP equations for the coloring model.We are now ready to describe in more detail our SSM analysis. Specifically, to get a boundon the norm k π − π ′ k , we will study the gradient of f c as we change the arguments ( π , . . . , π d )to ( π ′ , . . . , π ′ d ) along the line connecting them. Our gradient analysis will take account of thefollowing combinatorial notions. Definition 5.
A vertex v of T is called frozen under η if v ∈ Λ and non-frozen otherwise. Fora non-frozen vertex v of T , a color k is blocked for v (under η ) if there is a neighbor u ∈ Λ of v such that η ( u ) = k ; the color is called available for v otherwise. Observation 6.
In the setting of Theorem 4, we have that the disagreements between η and η ′ occur at distance at least 3 from the root. It follows that the set of the root’s children that arefrozen as well as the set of blocked colors for each of the non-frozen children are identical underboth η and η ′ . We will utilize that the gradient components that correspond to either frozen children orblocked colors can be disregarded since, by Observation 6, the corresponding arguments in (3)are fixed to the same value. Namely, we will track, for each color c , the fraction of non-frozenchildren which have color c available. This will allow us in the upcoming Lemma 10 to aggregateaccurately the gradient components corresponding to color c . The following definitions setupsome relevant notation. Definition 7.
Let D be the indices of the children of the root which are non-frozen under η and η ′ . For a color c ∈ [ q ] , let γ c ∈ [0 , be the fraction of indices i ∈ D such that color c isavailable for v i under η and η ′ (cf. Observation 6). Let γ and √ γ be the q -dimensional vectorwith entries { γ c } c ∈ [ q ] and {√ γ c } c ∈ [ q ] , respectively. Intuitively, if γ c is close to 0, color c is blocked at a lot of the children and hence the distance k π − π ′ k at the root should not depend a lot on the color c (since most components of thegradient corresponding to color c are zero).The following couple of definitions will be relevant for capturing more precisely the gradientof the functions { f c } c ∈ [ q ] . To begin with, the gradient will actually turn out to be related tothe value of f c as we move along the line ( π , . . . , π d ) to ( π ′ , . . . , π ′ d ). More precisely, we havethe following definition. Definition 8.
For t ∈ [0 , , let ˆ π ( t ) = { ˆ π c ( t ) } c ∈ [ q ] be the q -dimensional probability vector whose c -th entry is given by f c (cid:0) t π + (1 − t ) π ′ , . . . , t π d + (1 − t ) π ′ d (cid:1) . Note that ˆ π (1) = π and ˆ π (0) = π ′ ; in this sense, we can think of the vector ˆ π ( t ) as havingthe marginals at the root as we interpolate between ( π , . . . , π d ) to ( π ′ , . . . , π ′ d ).The next definition will be relevant for bounding the L norm of the gradient along the lineconnecting to ( π , . . . , π d ) to ( π ′ , . . . , π ′ d ). The bound will be in terms of the “marginals” atthe root, as captured by the vector ˆ π ( t ) (cf. Definition 8), and the availability of the q colors atthe children, as captured by the vector γ (cf. Definition 7). In particular, we will be interestedin the L norm of the following matrix, which is an idealized version to the Jacobian of the BPequation (see (13) for the precise formula). For a square matrix M , we use k M k to denote its L norm, i.e., k M k = max k x k =1 k Mx k . A fact thatwill be useful later is that k M k = max k x k =1 k x ⊺ M k , even for non-symmetric matrices M . efinition 9. Let ˆ π , ˆ γ be q -dimensional vectors with non-negative entries. The matrix M ˆ π , ˆ γ corresponding to the vectors ˆ π , ˆ γ is given by (cid:0) diag( ˆ π ) − ˆ π ˆ π ⊺ (cid:1) diag(ˆ γ (cid:1) . Our first main lemma shows how to bound the distance between the marginals at the rootunder η and η ′ , i.e., (cid:13)(cid:13) π − π ′ (cid:13)(cid:13) , in terms of the aggregate distance at the children. The newingredient in our bound is to account more carefully for the availability of the colors at thechildren (i.e., the vector γ ). Lemma 10.
Let q, d be positive integers so that q ≥ d + 2 . Then (cid:13)(cid:13) π − π ′ (cid:13)(cid:13) ≤ | D | K X i ∈ [ d ] (cid:13)(cid:13) π i − π i ′ (cid:13)(cid:13) where K := − q − d max t ∈ (0 , (cid:13)(cid:13)(cid:13) M ˆ π ( t ) , √ γ (cid:13)(cid:13)(cid:13) ,where D, γ , √ γ are as in Definition 7, ˆ π ( t ) is as in Definition 8, and M ˆ π ( t ) , √ γ is as in Defini-tion 9. Given Lemma 10, we are left with obtaining a good upper bound on the norm (cid:13)(cid:13) M ˆ π ( t ) , √ γ (cid:13)(cid:13) that takes advantage of the presence of the vector γ . It is not hard to see that the L norm ofthe matrix (cid:0) diag( ˆ π ) − ˆ π ˆ π ⊺ (cid:1) is bounded by max j ∈ [ q ] ˆ π j . The following result can be seen as ageneralisation of this fact, which is however significantly more involved to prove. The proof isgiven in Section 4. Lemma 11.
Let q be a positive integer, ˆ π be a q -dimensional probability vector and ˆ γ be a q -dimensional vector with non-negative entries which are all bounded by 1. Then, the L normof the matrix M ˆ π , ˆ γ = (cid:0) diag( ˆ π ) − ˆ π ˆ π ⊺ (cid:1) diag(ˆ γ (cid:1) satisfies (cid:13)(cid:13) M ˆ π , ˆ γ (cid:13)(cid:13) ≤
12 max j ∈ [ q ] ˆ π j (cid:0) γ j ) (cid:1) , where { ˆ π j } j ∈ [ q ] , { ˆ γ j } j ∈ [ q ] are the entries of ˆ π , ˆ γ , respectively. The final component of our proof is to utilize the bound in Lemma 11 to derive an upperbound on the norm of the matrix M ˆ π ( t ) , √ γ appearing in Lemma 10. To prove Theorem 4, weroughly need to show that the norm is bounded by 1 / | D | . We show that this is indeed the casein Section 6. Lemma 12.
There exist absolute constants β > and K ′ ∈ (0 , such that the following holdsfor all positive integers q, d satisfying q ≥ . d + β .Let γ , ˆ π ( t ) be the q -dimensional vectors of Definitions 7 and 8, respectively. Then, for all t ∈ [0 , and all colors k ∈ [ q ] , it holds that
12 ˆ π k ( t )(1 + γ k ) < K ′ / | D | , where D is the set of non-frozen children of ρ under η and η ′ . Assuming Lemmas 10, 11 and 12 for now, we next conclude the proof of Theorem 4.
Proof of Theorem 4.
Let U ′ := (1 + K ′ ) / K ′ ∈ (0 ,
1) is the constant in Lemma 12. Let β > q ≥ . d + β , the conclusion of Lemma 12applies and − q − d ) K ′ < U ′ . We will show that (cid:13)(cid:13) π − π ′ (cid:13)(cid:13) ≤ U max i ∈ [ d ] (cid:13)(cid:13) π i − π i ′ (cid:13)(cid:13) , with U := ( U ′ ) . (4) For a vector x , diag( x ) denotes the diagonal matrix with the entries of x on the diagonal. (cid:13)(cid:13) π − π ′ (cid:13)(cid:13) ≤ U | D | X i ∈ [ d ] (cid:13)(cid:13) π i − π ′ i (cid:13)(cid:13) . Note that an index i / ∈ D corresponds to a frozen child v i and therefore π i = π ′ i for all i / ∈ D and hence 1 | D | X i ∈ [ d ] (cid:13)(cid:13) π i − π ′ i (cid:13)(cid:13) ≤ max i ∈ [ d ] (cid:13)(cid:13) π i − π i ′ (cid:13)(cid:13) , proving (4). This completes the proof of Theorem 4. In this section, we prove Lemma 11.
Proof of Lemma 11.
For this proof, it will be convenient to simplify notation and use π in-stead of ˆ π and γ instead of ˆ γ , so that M π , γ becomes (cid:0) diag( π ) − ππ ⊺ (cid:1) diag( γ (cid:1) . Let C := max j ∈ [ q ] π j (1 + γ j ). We will establish that k M π , γ k ≤ C by showing that for an arbitrary q -dimensional vector x it holds that k x ⊺ M π , γ k ≤ C k x k . (5)We will focus on proving (5) in the case where the entries of the vector γ are all nonnegativeand strictly less than one; the case where some of the entries of γ are equal to 1 follows fromthe continuity of (5) with respect to γ .So, assume that γ j ∈ [0 ,
1) for all j ∈ [ q ]. Observe that k x ⊺ M π , γ k = X j ∈ [ q ] π j γ j ( x j − w ) where w := X j ∈ [ q ] π j x j . Let y j = x j − w for j ∈ [ q ]. Since π is a probability vector, we have X j ∈ [ q ] π j y j = 0 . Moreover, we can rewrite (5) as X j ∈ [ q ] π j γ j C y j ≤ X j ∈ [ q ] ( y j + w ) . (6)Note that the function f ( z ) = P j ∈ [ q ] ( y j + z ) achieves its minimum for z ∗ = − q P j ∈ [ q ] y j and f ( z ∗ ) = P j ∈ [ q ] y j − q (cid:0) P j ∈ [ q ] y j (cid:1) . Hence, to prove (6) (and therefore (5)), it suffices to showthat (cid:18) X j ∈ [ q ] y j (cid:19) ≤ q X j ∈ [ q ] y j A j , where A j := C C − π j γ j (7)Note that the A j ’s are well-defined and greater than 1 for all j ∈ [ q ] by our assumption that γ j ∈ [0 , P j ∈ [ q ] π j y j = 0, we therefore obtain that(7) is equivalent to (cid:18) X j ∈ [ q ] y j (1 + tπ j ) (cid:19) ≤ q X j ∈ [ q ] y j A j , where A j := C C − π j γ j , (8)7or any real number t — we will specify t soon (cf. the upcoming (10)). In particular, by theCauchy-Schwarz inequality, we have (cid:18) X j ∈ [ q ] y j (1 + tπ j ) (cid:19) ≤ X j ∈ [ q ] y j A j X j ∈ [ q ] A j (1 + tπ j ) , so (8) and hence (7) will follow if we find t such that X j ∈ [ q ] A j (1 + tπ j ) ≤ q. (9)We will choose t to minimise the l.h.s. in (9), i.e., set t := − P j ∈ [ q ] A j π j P j ∈ [ q ] A j π j , so that X j ∈ [ q ] A j (1 + tπ j ) = X j ∈ [ q ] A j − (cid:0) P j ∈ [ q ] A j π j (cid:1) P j ∈ [ q ] A j π j . (10)Therefore, for this choice of t , (9) becomes X j ∈ [ q ] ( A j − X j ∈ [ q ] A j π j ≤ (cid:18) X j ∈ [ q ] A j π j (cid:19) . (11)Using that A j = C C − π j γ j , (11) is equivalent to (note the division by C of both sides) X j ∈ [ q ] π j γ j C − π j γ j X j ∈ [ q ] π j C − π j γ j ≤ (cid:18) X j ∈ [ q ] Cπ j C − π j γ j (cid:19) . (12)We next establish (12). We can upper bound the l.h.s. of (12) using the inequality ab ≤ (cid:0) a + b (cid:1) ,which gives that X j ∈ [ q ] π j γ j C − π j γ j X j ∈ [ q ] π j C − π j γ j ≤ (cid:18) X j ∈ [ q ] π j (1 + γ j )2( C − π j γ j ) (cid:19) . So, to prove (12), it suffices to show that for each i ∈ [ q ], it holds that π j (1 + γ j )2( C − π j γ j ) ≤ Cπ j C − π j γ j which is indeed true, since C ≥ π j (1 + γ j ) for all i ∈ [ q ] by the definition of C .This proves (12), which in turn establishes (8) for the choice of t in (10). This yields (7)and hence (5) as well, finishing the proof of Lemma 11. In this section, we prove Lemma 10.
Proof of Lemma 10.
For i ∈ [ d ] and j ∈ [ q ], let F ( i ) c,j ( x ) be the partial derivative ∂f c ∂x i,j viewedas a function of the “concatenated” vector x = ( x , . . . , x d ). Note that, whenever x i,j = 1, wehave that F ( i ) c,j ( x ) = − f c ( x , . . . , x d ) − (cid:0) f c ( x , . . . , x d )) − x i,j if j = c,F ( i ) c,j ( x ) = f c ( x , . . . , x d ) f j ( x , . . . , x d )1 − x i,j if j = c. (13)8s mentioned earlier, we will interpolate between π and π ′ by interpolating along the straight-line segment connecting ( π , . . . , π d ) and ( π ′ , . . . , π ′ d ). In particular, for t ∈ [0 , π c ( t )denote the c -th entry of the vector ˆ π ( t ) defined in the statement of the lemma. Then, we havethat ˆ π c ( t ) = f c ( z ( t )) , where z ( t ) is the vector (cid:0) t π + (1 − t ) π ′ , . . . , t π d + (1 − t ) π ′ d (cid:1) . (14)We will use z i,j ( t ) to denote the j -th entry of the i -th vector in z ( t ), i.e., z i,j ( t ) = tπ i,j +(1 − t ) π ′ i,j .Let D be the set of indices i such that v i is not frozen under η and η ′ (cf. Observation 6).Observe that, for all i / ∈ D and c, j ∈ [ q ], we have that z i,j ( t ) = π i,j = π ′ i,j for t ∈ [0 , i ∈ D and j ∈ [ q ] we have that π i,j , π ′ i,j ≤ / ( q − d ) (since the child v i has at least q − d available colors in the subtree T i ) and hence0 ≤ z i,j ( t ) ≤ / ( q − d ) . (15)Since z i,j ( t ) = 1 for i ∈ D and j ∈ [ q ], it follows that d ˆ π c dt = X i ∈ D q X j =1 F ( i ) c,j ( z ( t ))( π i,j − π ′ i,j ) . Using (3), we therefore have that( π c − π ′ c ) = (cid:0) ˆ π c (1) − ˆ π c (0) (cid:1) = (cid:16) Z d ˆ π c dt dt (cid:17) = (cid:16) Z X i ∈ D q X j =1 F ( i ) c,j ( z ( t ))( π i,j − π ′ i,j ) dt (cid:17) ≤ Z (cid:18) X i ∈ D q X j =1 F ( i ) c,j ( z ( t ))( π i,j − π ′ i,j ) (cid:19) dt, where the last inequality follows by applying the Cauchy-Schwarz inequality for integrals. Bysumming over all colors c ∈ [ q ], we obtain (cid:13)(cid:13) π − π ′ (cid:13)(cid:13) ≤ Z q X c =1 (cid:18) X i ∈ D q X j =1 F ( i ) c,j ( z ( t ))( π i,j − π ′ i,j ) (cid:19) dt. (16)To simplify the r.h.s. of (16), we first note that, by (13) and (14), we have F ( i ) c,j ( z ( t )) = A c,j ( t )1 − z i,j ( t ) where A c,j := ( (cid:0) ˆ π c ( t )) − ˆ π c ( t ) , if j = c, ˆ π c ( t )ˆ π j ( t ) , if j = c (17)Moreover, for j ∈ [ q ], set u j ( t ) = 1 | D | γ j X i ∈ D π i,j − π ′ i,j − z i,j ( t ) if γ j >
0, else set u j ( t ) = 0 . (18)Note that if color j is blocked for the child v i we have that π i,j − π ′ i,j = 0, so using the powermean inequality we have that γ j ( u j ( t )) ≤ | D | X i ∈ D (cid:16) π i,j − π ′ i,j − z i,j (cid:17) . (19)Then, for c ∈ [ q ], we have that X i ∈ D q X j =1 F ( i ) c,j ( z ( t ))( π i,j − π ′ i,j ) = q X j =1 A c,j ( t ) X i ∈ D π i,j − π ′ i,j − z i,j ( t ) = | D | q X j =1 A c,j ( t ) γ j u j ( t ) , (20)9here the last equality follows from (18) and observing that if γ j = 0 then π i,j − π ′ i,j = 0 for all i ∈ D . Note that the ( c, q )-entry of M ˆ π ( t ) , √ γ is exactly − A c,j ( t ) √ γ j (cf. (17) and Definition 9)and hence, using (20), we can write the integrand in the r.h.s. of (16) as q X c =1 (cid:18) X i ∈ D q X j =1 F ( i ) c,j ( z ( t ))( π i,j − π ′ i,j ) (cid:19) = | D | (cid:13)(cid:13)(cid:13) M ˆ π ( t ) , √ γ u ( t ) (cid:13)(cid:13)(cid:13) , (21)where, for t ∈ [0 , u ( t ) is the q -dimensional vector with entries {√ γ j u j ( t ) } j ∈ [ q ] . Let W := max t ∈ [0 , (cid:13)(cid:13)(cid:13) M ˆ π ( t ) , √ γ (cid:13)(cid:13)(cid:13) , so that K = W − q − d . Then, for t ∈ [0 , (cid:13)(cid:13)(cid:13) M ˆ π ( t ) , √ γ u ( t ) (cid:13)(cid:13)(cid:13) ≤ W k u ( t ) k = W X j ∈ [ q ] γ j ( u j ( t )) ≤ W | D | X j ∈ [ q ] X i ∈ D (cid:13)(cid:13)(cid:13)(cid:13) π i,j − π ′ i,j − z i,j ( t ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ K | D | X j ∈ [ q ] X i ∈ D (cid:13)(cid:13) π i,j − π ′ i,j (cid:13)(cid:13) = K | D | X i ∈ [ d ] (cid:13)(cid:13) π i − π ′ i (cid:13)(cid:13) , (22)where the first inequality is by definition of the norm, the second inequality follows from (19),the third inequality follows from 0 ≤ z i,j ( t ) ≤ / ( q − d ), and the last equality follows from thefact that for i / ∈ D we have that π i = π ′ i . Combining (16), (21) and (22), we obtain that (cid:13)(cid:13) π − π ′ (cid:13)(cid:13) ≤ | D | K X i ∈ [ d ] (cid:13)(cid:13) π i − π ′ i (cid:13)(cid:13) . This finishes the proof of Lemma 10.
In this section, we prove Lemma 12. We begin with the following lemma.
Lemma 13.
Let q, d, h be positive integers so that q ≥ d + 1 and h ≥ . Let T = T d,h,ρ be the d -ary tree with height h rooted at ρ , Λ be a subset of the vertices of T such that ρ / ∈ Λ , and η : Λ → [ q ] be an extendible assignment of T . Then, for all colors k ∈ [ q ] that are available for ρ under η , it holds that µ T ( σ ρ = k | σ Λ = η ) ≥ (cid:0) − q − d (cid:1) d d + ( q − d ) (cid:0) − q − d (cid:1) d . Proof.
Let Q ⊆ [ q ] be the set of all colors that are available for ρ under η and let k ∈ Q . Let v , . . . , v d be the children of ρ in T and let D = { i ∈ [ d ] | v i / ∈ Λ } be the indices of the childrenof ρ that do not belong to Λ.For i ∈ [ d ], let T i = ( V i , E i ) be the subtree of T rooted at v i which consists of all descendantsof v i in T (together with v i itself). Further, for a color j ∈ [ q ], let x i,j = µ T i (cid:0) σ v i = j | σ Λ ∩ V i = η Λ ∩ V i (cid:1) , i.e., x i,j is the marginal probability that v i takes the color j at v i in µ T i with boundary condition η Λ ∩ V i . Note that0 ≤ x i,j ≤ q − d for all i ∈ D and j ∈ [ q ] , X j ∈ [ q ] x i,j = 1 for all j ∈ [ q ] . (23)10sing the tree recursion (2) and ignoring summands that are 0 or factors that are equal to 1,the marginal µ T ( σ ρ = k | σ Λ = η ) is expressed in terms of x i,j as follows: µ T ( σ ρ = k | σ Λ = η ) = Q i ∈ D (1 − x i,k ) P j ∈ Q Q i ∈ D (1 − x i,j ) . (24)We prove the lemma by deriving an appropriate lower bound on the quantity at the r.h.s. of(24) subject to the constraint in (23). For the numerator in (24), we have that Y i ∈ D (1 − x i,k ) ≥ (cid:18) − q − d (cid:19) | D | . (25)For the denominator we are going to show the following: X j ∈ Q Y i ∈ D (1 − x i,j ) ≤ d + ( q − d ) (cid:18) − q − d (cid:19) | D | . (26)Before showing that (26) is indeed true, note that the lemma follows by plugging (25), (26) into(24), yielding µ T ( σ ρ = k | σ Λ = η ) ≥ (cid:16) − q − d (cid:17) | D | d + ( q − d ) (cid:16) − q − d (cid:17) | D | ≥ (cid:16) − q − d (cid:17) d d + ( q − d ) (cid:16) − q − d (cid:17) d , where the last inequality follows by noting that the ratio in the middle is decreasing in | D | and | D | ≤ d .We now proceed with the proof of (26). First, we have the simple bound X j ∈ Q Y i ∈ D (1 − x i,j ) ≤ X j ∈ [ q ] Y i ∈ D (1 − x i,j ) . (27)For j ∈ [ q ], let x j = | D | P i ∈ D x i,j and note that ( x , . . . , x q ) is a probability vector whose entriesare in [0 , / ( q − d )]. By the AM-GM inequality, we can bound the r.h.s. of (27) by X j ∈ [ q ] Y i ∈ D (1 − x i,j ) ≤ X j ∈ [ q ] (1 − x j ) | D | . (28)It remains to observe that the function f ( z ) = P j ∈ [ q ] (1 − z j ) | D | is convex over the space ofprobability vectors z = ( z , . . . , z q ) whose entries are in [0 , / ( q − d )], and hence f attains itsmaximum at the extreme points of the space, which are given by (the permutations of) theprobability vector whose first d entries are equal to zero and the rest are equal to 1 / ( q − d ). Itfollows that X j ∈ [ q ] (1 − x j ) | D | ≤ d + ( q − d ) (cid:16) − q − d (cid:17) | D | . (29)Combining (27), (28) and (29) yields (26), thus concluding the proof of Lemma 13.We are now ready to prove Lemma 12. Proof of Lemma 12.
For convenience, let r = 1 .
59, so that q/d ≥ r . We will use that r satisfies C := 1 r exp (cid:16) r (cid:17) exp (cid:16) − r − (cid:0) r − (cid:1) (cid:17) < . (30)11e will show the result with the constant K ′ = (1 + C ) /
2. For the rest of this proof, we willfocus on the case q ∈ [1 . d + β, . d ], for some large constant β > q > . d thedesired bound follows rather crudely, see Footnote 4 below for details).Recall that v , . . . , v d are the children of ρ in T and D is the set of (indices of the) non-frozenchildren of the root ρ . Let Q ⊆ [ q ] be the set of all colors that are available for ρ under η ; sinceat most d − | D | colors can be blocked for ρ , we have that | Q | ≥ q − ( d − | D | ) . (31)For i ∈ [ d ], let T i = ( V i , E i ) be the subtree of T rooted at v i which consists of all descendantsof v i in T (together with v i itself). Further, for a color j ∈ [ q ], recall that π i,j = µ T i (cid:0) σ v i = j | σ Λ ∩ V i = η Λ ∩ V i (cid:1) ,π ′ i,j = µ T i (cid:0) σ v i = j | σ Λ ∩ V i = η ′ Λ ∩ V i (cid:1) , (32)i.e., π i,j is the marginal probability that v i takes the color j at v i in µ T i with boundary condition η Λ ∩ V i . For a non-frozen child v i (i.e., i ∈ D ), note that, if color j is available for v i (in T i ), thenwe have from Lemma 13 the bounds L ≤ π i,j , π i,j , where L = (cid:0) − q − d (cid:1) d d + ( q − d ) (cid:0) − q − d (cid:1) d . (33)Another useful bound to observe for later is that dL < / d ≥ k ∈ Q . For t ∈ [0 , z ( t ) be the vector (cid:0) t π + (1 − t ) π ′ , . . . , t π d +(1 − t ) π ′ d (cid:1) . Using the tree recursion (2) and ignoring summands that are 0 or factors that areequal to 1, we obtain ˆ π k ( t ) = Q i ∈ D (1 − z i,k ( t )) P j ∈ Q Q i ∈ D (1 − z i,j ( t )) . (34)Recall, our goal is to show that ˆ π k ( t )(1 + γ k ) < K ′ / | D | for all t ∈ [0 , γ k ∈ [0 ,
1] isthe fraction of non-frozen children that have color k available. Note that, if color j is availablefor the child v i , (33) gives that L ≤ z i,j ( t ) for t ∈ [0 , , so, using the fact that the color k is available for | D | γ k non-frozen children, we obtain that thenumerator of (34) is bounded by Y i ∈ D (cid:0) − z i,k ( t ) (cid:1) ≤ (1 − L ) | D | γ k ≤ exp( − L | D | γ k ) , (35)whereas the denominator, using the AM-GM inequality analogously to [7, Lemma 2.1 & Corol-lary 2.2], by X j ∈ Q Y i ∈ D (cid:0) − z i,j ( t ) (cid:1) = X j ∈ [ q ] Y i ∈ D (cid:0) − z i,j ( t ) (cid:1) − X j ∈ [ q ] \ Q Y i ∈ D (cid:0) − z i,j ( t ) (cid:1) ≥ (cid:0) q exp( −| D | /q ) − τ ) − ( d − | D | ) , (36)where τ > q, d, β ). From (34), (35), and (36), itfollows that ˆ π k ( t ) ≤ exp( − L | D | γ k ) q exp( −| D | /q ) − ( d −| D | ) − τ . Therefore, the lemma will follow by showing that | D | exp( −| D | Lγ k ) q exp( −| D | /q ) − ( d − | D | ) − τ (1 + γ k ) < K ′ . (37) For q > . d , we have from (34) and (31) that ˆ π k ( t ) ≤ | Q |−| D | ≤ q − d < . d ≤ K ′ / | D | , yielding the desiredinequality. h ( x ) = (1 + x ) exp( − dLx ) is increasing when x ∈ [0 , h ′ ( x ) = exp( − dLx ) (cid:0) − dL (1 + x ) (cid:1) ≥ exp( − dLx )(1 − dL ) > . Therefore, to prove (37), it suffices to show that | D | exp( −| D | L ) q exp( −| D | /q ) − ( d − | D | ) − τ < K ′ , or equivalently that f ( | D | ) > f ( x ) := K ′ (cid:0) q exp( − x/q ) − d + x − τ (cid:1) − x exp( − Lx ) for x ∈ [0 , d ]. We claim that f ( x ) isdecreasing in x . We have f ′ ( x ) = K ′ − K ′ exp( − x/q ) − exp( − Lx )(1 − Lx )which is maximised for x = d . In particular, f ′ ( x ) ≤ f ′ ( d ) = K ′ − K ′ exp( − d/q ) − exp( − dL )(1 − dL ) ≤ K ′ − K ′ exp( − /r ) − exp( − / − / ≤ , where the second to last inequality follows from the fact that dL < / K ′ <
1. For | D | = d , (38) becomes d exp( − dL ) q exp( − d/q ) − τ < K ′ . (39)Now, we have that dL ≥ r − (cid:0) d ( r − d − (cid:1) . Therefore, by choosing β large enough and using that q ∈ [1 . d + β, . d ], we can ensure that d exp( − dL ) q exp( − d/q ) − τ < C K ′ , where C is the constant in (30). This proves (39) and therefore concludes the proof of Lemma 12. Finally, utilizing Theorem 4, we give the proof of Theorem 3.
Proof of Theorem 3.
From Theorem 4, we know that there exist constants β > U ∈ (0 , q ≥ . d + β the conclusion of Theorem 4 applies. Note that Theorem 4 appliesto the d -ary tree rather than the ( d + 1)-regular tree but these trees differ only at the degree ofthe root. To account for it, we will assume that q ≥ . d + 1) + β , i.e., prove Theorem 3 withconstant β ′ = β + 1 .
59. Consider the function ζ given by ζ ( ℓ ) = 2 U ℓ − for ℓ ≥ ζ is exponentially decaying. We will show that the q -coloring model has strong spatial mixingon the ( d + 1)-regular tree with decay rate ζ .We first show by induction on h that, for the tree T = ˆ T d +1 ,h,ρ (that is, the ( d + 1)-ary treewith height h rooted at ρ ), for any subset Λ of vertices of T and arbitrary extendible assignments η, η ′ : Λ → [ q ] of T , it holds that (cid:13)(cid:13) π T,ρ,η − π T,ρ,η ′ (cid:13)(cid:13) ≤ ζ (dist( ρ, ∆)) , (40)where ∆ ⊆ Λ is the set of vertices where η and η ′ disagree. The base cases h = 0 , , h ≥ ℓ = dist( ρ, ∆). Once again, (40) is trivial when ℓ ≤ ℓ ≥ v , . . . , v d +1 be the children of ρ and, for i ∈ [ d + 1], let T i = ( V i , E i ) be the subtree of T rooted at v i which consists of all descendants of v i in T . Further,let π i = π T i ,v i ,η (Λ ∩ V i ) , π ′ i = π T i ,v i ,η ′ (Λ ∩ V i ) . Then, by Theorem 4 and since q ≥ . d + 1) + β ,we have that (cid:13)(cid:13) π T,ρ,η − π T,ρ,η ′ (cid:13)(cid:13) ≤ U max i ∈ [ d +1] (cid:13)(cid:13) π i − π i ′ (cid:13)(cid:13) . (41)For i ∈ [ d + 1], since T i is isomorphic to ˆ T d +1 ,h − ,ρ we have by the induction hypothesis that (cid:13)(cid:13) π i − π i ′ (cid:13)(cid:13) ≤ ζ ( ℓ − . Combining this with (41) and the fact that ζ ( ℓ ) = U ζ ( ℓ −
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