Local convergence of random graph colorings

Let $G=G(n,m)$ be a random graph whose average degree $d=2m/n$ is below the $k$-colorability threshold. If we sample a $k$-coloring $\sigma$ of $G$ uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart? According to a prediction from statistical physics, for average degrees below the so-called {\em condensation threshold} $d_c(k)$, the colors assigned to far away vertices are asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences 2007]. We prove this conjecture for $k$ exceeding a certain constant $k_0$. More generally, we investigate the joint distribution of the $k$-colorings that $\sigma$ induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem.


Introduction and results
Let G = G(n, m) denote the random graph on the vertex set [n] = {1, . . . , n} with precisely m edges. Unless specified otherwise, we assume that m = m(n) = dn/2 for a fixed number d > 0. As usual, G(n, m) has a property A "with high probability" ("w.h.p.") if lim n→∞ P [G(n, m) ∈ A] = 1.

Background and motivation
Going back to the seminal paper of Erdős and Rényi [22] that founded the theory of random graphs, the problem of coloring G(n, m) remains one of the longest-standing challenges in probabilistic combinatorics. Over the past half-century, efforts have been devoted to determining the likely value of the chromatic number χ(G(n, m)) [4,12,29,31] and its concentration [6,30,41] as well as to algorithmic problems such as constructing or sampling colorings of the random graph [3,16,18,19,24,26]. The single most tantalising feature of the random graph coloring problem is the interplay between local and global effects. Locally around almost any vertex the random graph is bipartite w.h.p. In fact, for any fixed average degree d > 0 and for any fixed ω the depth-ω neighborhood of all but o(n) vertices is just a tree w.h.p. Yet globally the chromatic number of the random graph may be large. Indeed, for any number k ≥ 3 of colors there exists a sharp threshold sequence d k-col = d k-col (n) such that for any fixed ε > 0, G(n, m) is k-colorable w.h.p. if 2m/n < d k-col (n) − ε, whereas the random graphs fails to be k-colorable w.h.p. if 2m/n > d k-col (n) + ε [1]. Whilst the thresholds d k-col are not known precisely, there are close upper and lower bounds. The best current ones read (1) where lim k→∞ δ k = lim k→∞ ε k = 0 [4,14,15]. To be precise, the lower bound in (1) is formally defined as (2) d k,cond = inf d > 0 : lim sup n→∞ E[Z k (G(n, m)) 1/n ] < k(1 − 1/k) d/2 . This number, called the condensation threshold due to a connection with statistical physics [27], can be computed precisely for k exceeding a certain constant k 0 [8]. An asymptotic expansion yields the expression in (1).
The contrast between local and global effects was famously pointed out by Erdős, who produced G(n, m) as an example of a graph that simultaneously has a high chromatic number and a high girth [21]. The present paper aims at a more precise understanding of this collusion between short-range and longrange effects. For instance, do global effects entail "invisible" constraints on the colorings of the local neighborhoods such that certain "local" colorings do not extend to a coloring of the entire graph? And what correlations do typically exist between the colors of vertices at a large distance?
Perhaps the most natural way of formalising these questions is as follows. Let k ≥ 3 be a number of colors, fix some number ω > 0 and assume that d < d k,cond so that G = G(n, m) is k-colorable w.h.p. Moreover, pick a vertex v 0 and fix a k-coloring σ 0 of its depth-ω neighborhood. How many ways are there to extend σ 0 to a k-coloring of the entire graph, and how does this number depend on σ 0 ? Additionally, if we pick another vertex v 1 that is "far away" from v 0 and if we pick another k-coloring σ 1 of the depth-ω neighborhood of v 1 , is there a k-coloring σ of G that simultaneously extends both σ 0 and σ 1 ? If so, how many such σ exist, and how does this depend on σ 0 , σ 1 ?
The main result of this paper (Theorem 1.1 below) provides a very neat and accurate answer to these questions. It shows that w.h.p. all "local" kcolorings σ 0 extend to asymptotically the same number of k-colorings of the entire graph. Let us write S k (G) for the set of all k-colorings of a graph G and let Z k (G) = |S k (G)| be the number of k-colorings. Moreover, let ∂ ω (G, v 0 ) be the depth-ω neighborhood of a vertex v 0 in G (i.e., the subgraph of G obtained by deleting all vertices at distance greater than ω from v 0 ). Then w.h.p. any k-coloring σ 0 of ∂ ω (G, v 0 ) has extensions to a k-coloring of G. Moreover, if we pick another vertex v 1 at random and fix some k-coloring σ 1 of the depth-ω neighborhood of v 1 , then w.h.p. the number of joint extensions of σ 0 , σ 1 is .
In other words, if we choose a k-coloring σ uniformly at random, then the distribution of the k-coloring that σ induces on the subgraph ∂ ω (G, v 0 ) ∪ ∂ ω (G, v 1 ), which is a forest w.h.p., is asymptotically uniform. The same statement extends to any fixed number v 0 , . . . , v l of vertices. This result, formally stated as Theorem 1.1/Corollary 1.2 below, is very much in line with and actually inspired by predictions from non-rigorous physics work. In fact, Corollary 1.3, a special case of Corollary 1.2, was conjectured explicitly in [27,44]. Moreover, also Theorem 1.1 and Corollary 1.2 hardly come as a surprise given the "replica symmetry breaking" picture drafted by physicists [25,27,44,32,33,34,35,36]. Furthermore, the results of this paper have a flavour of "decay of correlations" or "spatial mixing", a type of question that has been studied in prior work on sampling colorings of random graphs, e.g., [16,17,18,19,40]. However, we are not aware of a reference where Theorem 1.1 or Corollary 1.2 were conjecture explicitly.

Results
The appropriate formalism for describing the limiting behavior of the local structure of the random graph is the concept of local weak convergence [5,9]. The concrete installment of the formalism that we employ is reminiscent of that used in [11,38]. (Corollary 1.2 below provides a statement that is equivalent to the main result but that avoids the formalism of local weak convergence.) Let G be the set of all locally finite connected graphs whose vertex set is a countable subset of R. Further, let G k be the set of all triples (G, v 0 , σ) . Thus, ϕ preserves the root, the coloring and the order of the vertices (which are reals). Let [G, v 0 , σ] be the isomorphism class of (G, v 0 , σ) and let G k be the set of all isomorphism classes of rooted k-colored graphs.
For an integer ω ≥ 0 and Γ ∈ G k we let ∂ ω Γ denote the isomorphism class of the rooted k-colored graph obtained from Γ by deleting all vertices whose distance from the root exceeds ω. Then any Γ , ω ≥ 0 give rise to a function We endow G k with the coarsest topology that makes all of these functions continuous. Further, for l ≥ 1 we equip G l k with the corresponding product topology. Additionally, the set P(G l k ) of probability measures on G l k carries the weak topology, as does the set P 2 (G l k ) of all probability measures on P(G l k ). The spaces G l k , P(G l k ), P 2 (G l k ) are Polish [5]. For Γ ∈ G k we denote by δ Γ ∈ P(G k ) the Dirac measure that puts mass one on Γ .
Let G be a finite k-colorable graph whose vertex set V (G) is contained in R and let v 1 , . . . , v l ∈ V (G). Then we can define a probability measure on G l k as follows. Letting G v denote the connected component of v ∈ V (G) and σ v the restriction of σ : The idea is that λ G,v 1 ,...,v l captures the joint empirical distribution of colorings induced by a random coloring of G "locally" in the vicinity of the "roots" v 1 , . . . , v l . Further, let This measure captures the typical distribution of the local colorings in a random graph with l randomly chosen roots. We are going to determine the limit of λ l n,m,k as n → ∞. To characterise this limit, let T * (d) be a (possibly infinite) random Galton-Watson tree rooted at a vertex v * 0 with offspring distribution Po(d). We embed T * (d) into R by independently mapping each vertex to a uniformly random point in [0, 1]; with probability one, all vertices get mapped to distinct points. Let T (d) ∈ G signify the resulting random tree and let v 0 denote its root. For a number ω > 0 we let ∂ ω T (d) denote the (finite) rooted tree obtained from T (d) by removing all vertices at a distance greater than ω from v 0 . Moreover, for l ≥ 1 let T 1 (d), . . . , T l (d) be l independent copies of T (d) with roots v 1 0 , . . . , v l 0 and set (4)).
The sequence (ϑ l d,k [ω]) ω≥1 converges (see Appendix A) and we let Combinatorially, ϑ l d,k corresponds to sampling l copies of the Galton-Watson tree T (d) independently. These trees are colored by assigning a random color to each of the l roots independently and proceeding down each tree by independently choosing a color for each vertex from the k − 1 colors left unoccupied by the parent.
Fix numbers ω ≥ 1, l ≥ 1, choose a random graph G = G(n, m) for some large enough n and choose vertices v 1 , . . . , v l uniformly and independently at random. Then the depth-ω neighborhoods ∂ ω (G, v 1 ), . . . , ∂ ω (G, v l ) are pairwise disjoint and the union F = ∂ ω (G, v 1 ) ∪ · · · ∪ ∂ ω (G, v l ) is a forest w.h.p. Moreover, the distance between any two trees in F is Ω(ln n) w.h.p. Given that G is k-colorable, let σ be a random k-coloring of G. Then σ induces a k-coloring of the forest F. Theorem 1.1 implies that w.h.p. the distribution of the induced coloring is at a total variation distance o(1) from the uniform distribution on the set of all k-colorings of F. Formally, let us write µ k,G for the probability distribution on [k] V (G) defined by i.e., the uniform distribution on the set of k-colorings of the graph G. Moreover, for U ⊂ V (G) let µ k,G|U denote the projection of µ k,G onto [k] U , i.e., If H is a subgraph of G, then we just write µ k,G|H instead of µ k,G|V (H) . Let · TV denote the total variation norm.
Since w.h.p. the pairwise distance of l randomly chosen vertices v 1 , . . . , v l in G is Ω(ln n), we observe that w.h.p.
With very little work it can be verified that Corollary 1.2 is actually equivalent to Theorem 1.1. Setting ω = 0 in Corollary 1.2 yields the following statement, which is of interest in its own right.
There is a number k 0 > 0 such that for all k ≥ k 0 , d < d k,cond and any integer l > 0 we have By the symmetry of the colors, µ k,G|{v} is just the uniform distribution on [k] for every vertex v. Hence, Corollary 1.3 states that for d < d k,cond w.h.p. in the random graph G for randomly chosen vertices v 1 , . . . , v l the following is true: if we choose a k-coloring σ of G at random, then (σ(v 1 ), . . . , σ(v l )) ∈ [k] l is asymptotically uniformly distributed. Prior results of Montanari and Gershenfeld [23] and of Montanari, Restrepo and Tetali [39] imply that (6) holds for d < 2(k − 1) ln(k − 1), about an additive ln k below d k,cond .
The above results and their proofs are inspired by ideas from statistical physics. More specifically, physicists have developed a non-rigorous but analytic technique, the so-called "cavity method" [32], which has led to various conjectures on the random graph coloring problem. These include a prediction as to the precise value of d k,cond for any k ≥ 3 [44] as well as a conjecture as to the precise value of the k-colorability threshold d k-col [28]. While the latter formula is complicated, asymptotically we expect that d k-col = (2k − 1) ln k − 1 + ε k , where lim k→∞ ε k = 0. According to this conjecture, the upper bound in (1) is asymptotically tight and d k-col is strictly greater than d k,cond . Furthermore, according to the physics considerations (6) holds for any k ≥ 3 and any d < d k,cond [27]. Corollary 1.3 verifies this conjecture for k ≥ k 0 . By contrast, according to the physics predictions, (6) does not hold for d k,cond < d < d k-col . As (6) is the special case of ω = 0 of Theorem 1.1 (resp. Corollary 1.2), the conjecture implies that neither of these extend to d > d k,cond . In other words, the physics picture suggests that Theorem 1.1, Corollary 1.2 and Corollary 1.3 are optimal, except that the assumption k ≥ k 0 can possibly be replaced by k ≥ 3.

Remark 1.4.
The assumption k ≥ k 0 comes from the corresponding assumption in [8,15], which give no numerical clue as to the value of k 0 . Thus, the issue is that the condensation threshold is not known for small k. Yet for any k ≥ 3 the proofs of Theorem 1.1 goes through for d < 2(k−1) ln(k−1), which is the degree up to which the second moment argument from [4] succeeds.

An application
Suppose we draw a k-coloring σ of G at random and consider some vertex v. What is the effect of the assignment of v to the assignment of the rest of the vertices in the graph? Of course, σ assigns to the neighbors of v colors that are distinct to that of v. More generally, it seems reasonable to expect that for any fixed "radius" ω the colors assigned at v influences the color assignment of the vertices at distance ω from v. But do these correlations persist as ω → ∞? This is the core question of the so-called "reconstruction problem". The reconstruction problem has received considerable attention in the context of random constraint satisfaction problems in general and in random graph coloring in particular [27,23,39,20,10,42]. To illustrate the use of Theorem 1.1 we will show how it readily implies the result on the reconstruction problem for random graph coloring from [39].
The reconstruction problem considers the effect of the color assignment of a vertex v to the vertices at distance ω in a random coloring of G as ω → ∞ ("point-to-set correlation"). Equivalently, the problem can be formulated by considering the effect on the coloring of v by a "typical coloring" of the vertices at distance ω from v as ω → ∞. We formally state the problem by considering the second approach.
Assume that G is a finite k-colorable graph. For v ∈ V (G) and a subset ∅ = R ⊂ S k (G) let µ k,G|v ( · |R) be the probability distribution on [k] defined by i.e., the distribution of the color of v in a random coloring σ ∈ R. For v ∈ V (G), ω ≥ 1 and σ 0 ∈ S k (G) let Thus, R k,G (v, ω, σ 0 ) contains all k-colorings that coincide with σ 0 on vertices whose distance from v is at least ω. Moreover, let Clearly, for symmetry reasons, if we draw a k-coloring σ ∈ S k (G) uniformly at random, then σ(v) is uniformly distributed over [k]. What bias k,G (v, ω, σ 0 ) measures is how much conditioning on the event σ ∈ R k,G (v, ω, σ 0 ) biases the color of v. Accordingly, bias k,G (v, ω) measures the bias induced by a random "boundary condition" σ 0 . We say that we have non-reconstruction occurs for the k-colorings of G(n, m) if Otherwise, we have reconstruction.
Analogously, recalling that T (d) is the Galton-Watson tree rooted at v 0 , we say that tree non-reconstruction occurs at d if Otherwise, tree reconstruction occurs.
There is a number k 0 > 0 such that for all k ≥ k 0 and d < d k,cond the following is true.
(7) Reconstruction occurs in G(n, m) ⇔ tree reconstruction occurs at d.
Montanari, Restrepo and Tetali [39] proved (7) for d < 2(k − 1) ln(k − 1), about an additive ln k below d k,cond . This gap could be plugged by invoking recent results on the geometry of the set of k-colorings [7,14,37]. However, we shall see that Corollary 1.5 is actually an immediate consequence of Theorem 1.1.
The point of Corollary 1.5 is that it reduces the reconstruction problem on a combinatorially extremely intricate object, namely the random graph G(n, m), to the same problem on a much simpler structure, namely the Galton-Watson tree T (d). That said, the reconstruction problem on T (d) is far from trivial. The best current bounds show that there exists a sequence

Preliminaries and notation
be the number of edges of the complete graph on n vertices that are monochromatic under σ. Similarly, for two maps σ, τ : [n] → [k] let F(σ, τ ) be the number of edges of the complete graph that are monochromatic under either σ or τ . If we define the overlap of σ, τ : then by inclusion/exclusion we have We can view ρ(σ, τ ) as a distribution on [k] × [k]. Throughout the paper we Let G be a k-colorable graph. By σ k,G , σ k,G 1 , σ k,G 2 , . . . ∈ S k (G) we denote independent uniform samples from S k (G). Where G, k are apparent from the context, we omit the superscript. Moreover, if X : We omit the subscript G and/or k where it is apparent from the context. Thus, the symbol · G,k refers to the average over randomly chosen kcolorings of a fixed graph G. By contrast, the standard notation E [ · ], P [ · ] will be used to indicate that the expectation/probability is taken over the choice of the random graph. Unless specified otherwise, we use the standard O-notation to refer to the limit n → ∞. Throughout the paper, we tacitly assume that n is sufficiently large for our various estimates to hold.
By a rooted graph we mean a graph G together with a distinguished vertex v, the root. The vertex set is always assumed to be a subset of R. If ω ≥ 0 is an integer, then ∂ ω (G, v) signifies the subgraph of G obtained by removing all vertices at distance greater than ω from v (including those vertices of G that are not reachable from v), rooted at v. An isomorphism between two rooted graphs (G, v), (G , v ) is an isomorphism G → G of the underlying graphs that maps v to v and that preserves the order of the vertices (which is why we insist that they be reals).
For a finite or countable set X we denote by P(X ) the set of all probability distributions on X , which we identify with the set of all maps p : X → [0, 1] such that x∈X p(x) = 1. Furthermore, if N > 0 is an integer, then P N (X ) is the set of all p ∈ P(X ) such that N p(x) is an integer for every x ∈ X . With the convention that 0 ln 0 = 0, we denote the entropy of p ∈ P(X ) by Finally, we need the following inequality. Lemma 1.6 ( [43]). Let X 1 , . . . , X N be independent random variables with values in a finite set Λ. Assume that f : Λ N → R is a function, that Γ ⊂ Λ N is an event and that c, c > 0 are numbers such that the following is true.
Then for any γ ∈ (0, 1] and any t > 0 we have

Outline
None of the arguments in the present paper are particularly difficult. It is rather that a combination of several relatively simple ingredients proves quite powerful. In this section we give an overview of the various pieces and their interplay. In a nutshell, we are going to prove Theorem 1.1 and its corollaries by studying random pairs (σ 1 , σ 2 ) of k-colorings of the random graph G. Specifically, we are going to show that for any fixed integer ω ≥ 0, any fixed rooted tree T and any two k-colorings τ 1 , What might be called a subtle doublecounting argument then yields Theorem 1.1. This proof strategy can be viewed as a generalisation of the arguments from [23,39], which are based on studying the "vertex overlap" ρ(σ 1 , σ 2 ) of two random k-colorings of G rather than the aforementioned "tree overlaps".

The number of k-colorings
In order to study random pairs (σ 1 , σ 2 ) of k-colorings we employ a concentration result for the total number Z k (G) of k-colorings.
To put Theorem 2.1 to work we recall the formula for the first moment of the number of k-colorings.
Although Lemma 2.2 is folklore, let us briefly comment on how the expression comes about. For any σ : By convexity we have F(σ) ≥ 1 k n 2 − n for all σ. In combination with (11) and the linearity of expectation, this implies that Plugging the asymptotic formula (1) for d k,cond into Lemma 2.2, we find that E[Z k (G)] = exp(Ω(n)) for k > k 0 and d < d k,cond . Hence, Theorem 2.1 establishes a remarkably strong form of concentration: the random variable ln Z k (G), typically of order n, has bounded fluctuations.
As pointed out in [7], Theorem 2.1 gives us a handle on the experiment of first generating a random graph G and then sampling a single k-coloring σ of G uniformly at random. Namely, the distribution of the pair (G, σ) can be approximated by a much simpler probability distribution, the socalled "planted model". Indeed, the approximation enabled by Theorem 2.1 is much more accurate than the one previously established in [2]. However, by itself even the result from [7] is not powerful enough to derive Theorem 1.1 (cf. also the discussion in [11]). Instead, we are going to have to cope with the experiment of sampling a random pair (σ 1 , σ 2 ) of colorings of G.

Planting replicas
To this end, we consider a probability distribution π pr n,m,k on triples (G, σ 1 , σ 2 ) such that G is a graph on [n] with m edges and σ 1 , σ 2 ∈ S k (G): the planted replica model is induced by the following experiment.
PR1: Sample a pair (σ 1 ,σ 2 ) of maps [n] → [k] uniformly at random from the set of all pairs such that PR2: Choose a graphĜ on [n] with precisely m edges uniformly at random, subject to the condition that bothσ 1 ,σ 2 are proper k-colorings.
We define It is easy to bring the known techniques from the theory of random graphs to bear on the planted replica model. Indeed, the conditioning in PR1 is Moreover, PR2 just means that we draw m random edges out of the n 2 −F(σ 1 ,σ 2 ) edges of the complete graph that are bichromatic under bothσ 1 ,σ 2 . In particular, By contrast, the experiment of first choosing a random graph G and then sampling two k-colorings σ 1 , σ 2 uniformly at random seems far less amenable. Formally, the random replica model π rr n,m,k is the probability distribution on triples (G, σ 1 , σ 2 ) induced by the following experiment.
RR1: Choose a random graph G = G(n, m) subject to the condition that G is k-colorable. RR2: Sample two colorings σ 1 , σ 2 of G uniformly and independently.
Thus, the random replica model is defined by the formula (12) π rr n,m,k (G, σ 1 , If d < d k,cond , then G is k-colorable w.h.p. and thus the conditioning in RR1 is innocent. But this is far from true of the experiment described in RR2. For instance, we have no idea as to how one might implement RR2 efficiently for d anywhere near d k,cond . In fact, the best current algorithms for finding a single k-coloring of G, let alone a random pair, stop working for degrees d about a factor of two below d k,cond (cf. [2]).
Yet for d < d k,cond , the "difficult" random replica model can be studied by means of the "simple" planted replica model. More precisely, recall that a sequence (µ n ) n of probability measures is contiguous with respect to another sequence (ν n ) n if µ n , ν n are defined on the same ground set for all n and if for any sequence (A n ) n of events such that lim n→∞ ν n (A n ) = 0 we have lim n→∞ µ n (A n ) = 0. Proposition 2.3. There is k 0 such that for all k ≥ k 0 , d < d k,cond , π rr n,m,k is contiguous with respect to π pr n,m,k .
The proof of Proposition 2.3, based on Theorem 2.1, can be found in Section 3. Apart from the concentration of Z k (G(n, m)), the proof involves a study of the overlap of two randomly chosen colorings of G(n, m). The overlap was studied in prior work on reconstruction [23,39] in the case that d < 2(k − 1) ln(k − 1) via the second moment argument of Achlioptas and Naor [4]. To extend the study of the overlap to the whole range d ∈ (0, d k,cond ), we harness insights from the recent improvements [8,15] of [4].

Tree overlaps
As mentioned initially, we aim to understand the "dicolored neighborhood statistics" of the random graph G, i.e., the colorings that two random kcolorings σ 1 , σ 2 of G induce on the neighborhoods ∂ ω (G, v), v ∈ [n], for a fixed radius ω. Formally, if ϑa is a rooted tree, τ 1 , τ 2 ∈ S k (ϑa), ω ≥ 0 and if G is a k-colorable graph on [n] and σ 1 , σ 2 ∈ S k (G), we define In words, this is the probability that the depth-ω neighborhood of a random vertex v is isomorphic to ϑa and that the coloring of this neighborhood induced by σ j coincides with τ j for j = 1, 2. We are going to study the random variables Q ϑa,τ 1 ,τ 2 ,ω (G, σ 1 , σ 2 ) on the random replica model by way of the planted replica model. Set Proposition 2.4. Let k ≥ 3 and d > 0. Let ϑa be a rooted tree, τ 1 , τ 2 ∈ S k (ϑa) and ω ≥ 0. Then for (Ĝ,σ 1 ,σ 2 ) chosen from π pr n,m,k we have Although the proof is based on standard techniques, it requires a bit of work, see Section 4.

Double-counting and local weak convergence
Combining Proposition 2.4 with a double-counting argument that generalises an elegant idea from [23], we obtain the following.
In words, suppose we first choose a (k-colorable) random graph G, a random k-coloring σ of G and l random vertices v 1 , . . . , v l of G. Then n −l X n is the probability that for each i ∈ [l] the depth-ω neighborhood ∂ ω (G, v i ) is isomorphic to ϑa i and that the colorings that σ induces on ∂ ω (G, v i ) coincide with τ i for all i.
To get an idea of how Proposition 2.5 follows from Proposition 2.4, let us consider the simplest case. If l = 2, ω = 0, then we aim to show that w.h.p. for all but o(n 2 ) vertex pairs v 1 , v 2 , the random pair (σ(v 1 ), σ(v 2 )) ∈ [k] 2 is asymptotically uniformly distributed. (The following computations are along the lines of [23], which actually deals with the case l ≥ 2 and ω = 0.) Thus, fix any two colors The first equality sign holds because 1{σ(v 1 ) = i 1 } G = 1{σ(v 2 ) = i 2 } G = k −1 by symmetry. The second one is due to the independence of σ 1 , σ 2 .
Combining both these observations, we see that the last expression equals G . Further, applying Proposition 2.4 to the trees τ 1 , τ 2 consisting of the root only and invoking Proposition 2.3, we conclude that (14) is o(n 2 ). In fact, according to Proposition 2.4 w.h.p. in (Ĝ,σ 1 , Hence, by Proposition 2.3 the same is true of the triple (G, σ 1 , σ 2 ) w.h.p. and therefore A similar argument applies to the other three terms. Hence, retracing our steps, we obtain Thus, by Markov's inequality and Cauchy-Schwarz, w.h.p. we have Finally, taking a union bound over i 1 , i 2 ∈ [k], we conclude that w.h.p. for all but o(n 2 ) vertex pairs the distribution of (σ(v 1 ), σ(v 2 )) is within o(1) of the uniform distribution in total variation. The full proof of Proposition 2.5 can be found in Section 5. Theorem 1.1 follows rather immediately from Proposition 2.5; all that is required is unraveling the construction of the topology.
Proof of Theorem 1.1 (assuming Proposition 2.5). As P 2 (G l k ) carries the weak topology, we need to show that for any continuous f : P(G l k ) → R with a compact support, (15) lim Hence, there is ω 0 = ω 0 (ε) such that for ω > ω 0 we have Furthermore, the topology of G k is generated by the functions (3). Because f has a compact support, this implies that there is ω 1 = ω 1 (ε) such that for any ω > ω 1 (ε) and all Γ 1 , . . . , Γ l ∈ G k we have Hence, pick some ω > ω 0 + ω 1 and assume that n > n 0 (ε, ω) is large enough.
Let v 1 , . . . , v l denote vertices of G that are chosen independently and uniformly at random. By the linearity of expectation and the definitions of λ l n,m,k and λ G,v 1 ,...,v l ,
Let us call A, the quantity on the l.h.s. of the above equality. With G = G(n, m) it holds that We observe that, for any v-rooted G ∈ G and ω it holds that bias k,G (v, ω) ∈ [0, 1]. Then, by using Corollary 1.2 where l = 1 (i.e., weak convergence) we get that For bounding the second quantity we use the following observation: The above implies that where v * is a randomly chosen vertex of G. The probability term P [∂ ω (G, v * ) ∼ = ∂ ω T (d)] is w.r.t. any coupling of ∂ ω (G, v * ) and ∂ ω T (d). Also, the maximum index ϑa varies over all trees with at most n vertices and with at most ω levels.
Using the standard graph exploration process to obtain the depth-ω neighborhood of v * , there is a coupling of ∂ ω (G(n, m), v * ) and ∂ ω T (d), where d = 2m/n, such that Plugging (26) into (25) we get that since it always holds that bias k,ϑa (v, ω) ∈ [0, 1]. From (24) and (27) According to physics considerations, the deeper reason behind these issues is that the "shape" of the set of k-colorings changes at the condensation point. For d < d k,cond the set of k-colorings (provably) decomposes into tiny well-separated "clusters" that each carry mass exp(−Ω(n)) under the measure µ k,G w.h.p. [2,37]. However, for d k,cond < d < d k-col a bounded number of clusters are expected to contain a 1−o(1) fraction of the probability mass [27]. Hence, two randomly chosen k-colorings have a non-vanishing probability of belonging to the same cluster. In effect, the "vertex overlap" ρ(σ 1 , σ 2 ) is not concentrated aboutρ anymore, and thus even the weak decorrelation property (6) fails to hold. A detailed derivation of the physics predictions can be found in [32].
Nonetheless, the argument that we developed for d < d k,cond is reasonably generic, i.e., it does not depend much on the particulars of the k-colorability problem. We expect it to extend to alike "random constraint satisfaction problems" (say, with a similar "vertex overlap" behavior) up to their respective condensation thresholds. A natural class to think of are the binary problems studied in [39]. Another candidate might be the hardcore model, which was studied in [11] by a somewhat different approach.

The planted replica model
Throughout this section we assume that k ≥ k 0 for some large enough constant k 0 .
In this section we prove Proposition 2.3. A key step is to establish the following fact about the "vertex overlap". Lemma 3.1. Assume that d < d k,cond and let ω = ω(n) be such that lim n→∞ ω(n) = ∞. Then Thus, with high probability over the choice of the random graph G (the outer E) and over the choice of a pair of k-colorings (the inner · G ) the 2 -distance of the overlap fromρ is bounded by ω/n. The d < 2(k − 1) ln(k − 1) case of Lemma 3.1 was previously proved in [39] by way of the second moment analysis from [4]. As it turns out, the regime 2(k − 1) ln(k − 1) < d < d k,cond requires a somewhat more sophisticated argument. In any case, our proof of Lemma 3.1 below includes the case d < 2(k−1) ln(k−1), which does not add much to the argument.
Proof of Proposition 2.3. Assume for contradiction that (A n ) n≥1 is a sequence of events such that for some fixed number ε > 0 we have (52) lim n→∞ π pr n,m,k A n = 0 while lim sup n→∞ π rr n,m,k A n > 2ε.

Analysis of the planted replica model
In this section we assume that k ≥ 3 and that d > 0.
In this section we prove Proposition 2.4, which asserts that in the planted replica model, the distribution of the "dicoloring" thatσ 1 ,σ 2 induce in the depth-ω neighborhood of a random vertex v converges to the uniform distribution on the tree that the depth-ω neighborhood of v induces. The proof is by extension of an argument from [8] for the "standard" planted model (with a single coloring). More specifically, it is going to be convenient to work with the following binomial version π pr n,p,k of the planted replica model, where p ∈ (0, 1). PR1' sample two mapsσ 1 ,σ 2 : [n] → [k] independently and uniformly at random. PR2' generate a random graphG by including each of the n 2 − F(σ 1 ,σ 2 ) edges that are bichromatic under bothσ 1 ,σ 2 with probability p independently.
By construction, π pr n,p,k given A ∩ B is identical to π pr n,m,k given B. Consequently, (60) and (61)  The following proofs are based on a simple observation. Given the coloringsσ 1 ,σ 2 , we can constructG as follows. First, we simply insert each of the n 2 edges of the complete graph on [n] with probability p independently. The result of this is, clearly, the Erdős-Rényi random graph G(n, p). Then, we "reject" (i.e., remove) each edge of this graph that joins two vertices that have the same color under eitherσ 1 orσ 2 .  2. Let L be the event that there is a vertex v such that ∂ ω (G, v) contains more than n 0.1 vertices. Then π pr n,p,k [L] ≤ exp(−Ω(ln 2 n)).
Proof. The proof is based on Lemma 1.6. To apply it, we view (G,σ 1 ,σ 2 ) as chosen from a product space X 2 , . . . , and where X n+v is a 0/1 vector of length v−1 whose components are independent Be(p) variables for v ∈ [n]. Namely, represents the color pair (σ 1 (v),σ 2 (v)), and X n+v for v ∈ [n] indicates to which vertices w < v withσ 1 (w) =σ 1 (v),σ 2 (w) =σ 2 (v) vertex v is adjacent ("vertex exposure"). Define random variables S v = S v (G,σ 1 ,σ 2 ) and S by letting Then by (13) we have Furthermore, let G be the graph obtained fromG by removing all edges e that are incident with a vertex v such that |∂ ω G , v | > λ and let S v .
If Γ occurs, then S = S . Hence, (63) implies that The random variable S satisfies (10) with c = λ and c = n. Indeed, altering either the colors of one vertex u or its set of neighbors can only affect those vertices v that are at distance at most ω from u, and in G there are no more than λ such vertices. Thus, Lemma 1.6 applied with, say, t = n 2/3 and γ = 1/n and (63) yield Finally, the assertion follows from (62), (64) and (65).
The construction of this coupling is based on the principle of deferred decisions. More specifically, we are going to view the exploration of the depth-ω neighborhood of v * in the random graphG as a random process, reminiscent of the standard breadth-first search process for the exploration of the connected components of the random graph. The colors of the individual vertices and their neighbors are revealed in the course of the exploration process. The result of the exploration process will be a dicolored tree (T , u(v * ),τ 1 ,τ 1 ) whose vertex set is contained in [0, 1]. This tree is isomorphic to ∂ ω (G, v * ,σ 1 ,σ 2 ) w.h.p. Furthermore, the distribution of the tree is at total variance distance o(1) from that of (T (d), v 0 , τ 1 , τ 2 ).
Throughout the exploration process, every vertex is marked either dead, alive, rejected or unborn. The semantics of the marks is similar to the one in the usual "branching process" argument for the component exploration in the random graph: vertices whose neighbors have been explored are "dead", vertices that have been reached but whose neighbors have not yet been inspected are "alive", and vertices that the process has not yet discovered are "unborn". The additional mark "rejected" is necessary because we reveal the colors of the vertices as we explore them. More specifically, as we explore the neighbors of an alive v vertex, we insert a "candidate edge" between the alive vertex and every unborn vertex with probability p independently. If upon revealing the colors of the "candidate neighbor" w of v we find a conflict (i.e.,σ 1 (v) =σ 1 (w) orσ 2 (v) =σ 2 (w)), we "reject" w and the "candidate edge" {v, w} is discarded. Additionally, we will maintain for each vertex v a number D(v) ∈ [0, ∞]; the intention is that D(v) is the distance from the root v * in the part of the graph that has been explored so far. The formal description of the process is as follows.
The process stops once there is no alive vertex y such that D(y) < ω anymore, at which point we have got a treeT that is embedded into [0, 1]. Let A be the event that ∂ ω (Ĝ, v * ) is an acyclic subgraph that contains no more than n 0.1 vertices. Furthermore, let R be the event that in EX2 it never occurs that a vw = 1 for a rejected vertex w. Then Lemma 4.2 implies that Further, if A ∩ R occurs, then whenever EX2 processes an alive vertex v with D(v) < ω, the number of unborn neighbors of v of every color combination (s 1 , s 2 ) such that s 1 =σ(v), s 2 =σ(v) is a binomial random variable whose mean lies in the interval [(n − n 0.1 )p/k 2 , np/k 2 ]. The total variation distance of this binomial distribution and the Poisson distribution Po(d/(k − 1) 2 ), which is precisely the distribution of the number of children colored (s 1 , s 2 ) in the dicolored Galton-Watson tree, is O(n −0.9 ) by the choice of p. In addition, let B be the event that each interval ((i−1)/n, i/n) for i = 1, . . . , n contains at most one vertex of the tree ∂ ω T (d). Then P [B] = 1−o(1) and given A∩R and B, there is a coupling of (T , u(v * ),τ 1 ,τ 2 ) and ∂ ω (T (d), v 0 , τ 1 , τ 2 ) such that P ∂ ω (T (d), v 0 , τ 1 , τ 2 ) = (T , u(v * ),τ 1 ,τ 2 ) = 1 − o(1).
Proof. This follows by combining Lemmas 4.3 and 4.4.
Finally, Proposition 2.4 is immediate from Lemma 4.1 and Corollary 4.5.

Establishing local weak convergence
Throughout this section we assume that k ≥ k 0 for some large enough constant k 0 and that d < d k,cond .
In this section we prove Proposition 2.5. The purpose of Propositions 2.3 and 2.4 was to facilitate the proof of the following fact.
We complete the proof of Proposition 2.5 by generalising the elegant argument that was used in [23, Proposition 3.2] to establish a statement similar to the ω = 0 case of Proposition 2.5.