# Slow mixing of Glauber Dynamics for the hard-core model on regular bipartite graphs

###### Abstract

Let be a finite, -regular bipartite graph. For any let be the probability measure on the independent sets of in which the set is chosen with probability proportional to ( is the hard-core measure with activity on ). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is . We show that when is large enough (as a function of and the expansion of subsets of single-parity of ) then the convergence to stationarity is exponentially slow in . In particular, if is the -dimensional hypercube we show that for values of tending to as grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods.

^{†}

^{†}footnotetext: Key words: Mixing time, hard-core model, conductance, Glauber dynamics, discrete hypercube.

## 1 Introduction and statement of the result

Let be a simple, loopless, finite graph on vertex set and edge set . (For graph theory basics, see e.g. [3], [7].) Write for the set of independent sets (sets of vertices spanning no edges) in . For we define the hard-core measure with activity on by

(1) |

where is the appropriate normalizing constant. We will often write for and, for , for .

The hard-core measure originally arose in statistical physics (see e.g. [8, 1]) where it serves as a simple mathematical model of a gas with particles of non-negligible size. The vertices of we think of as sites that may or may not be occupied by particles; the rule of occupation is that adjacent sites may not be simultaneously occupied. The activity parameter measures the likelihood of a site being occupied.

The measure also has a natural interpretation in the context of communications networks (see e.g. [14]). Here the vertices of are thought of as locations from which “calls” can be made; when a call is made, the call location is connected to all its neighbours, and throughout its duration, no call may be placed from any of the neighbours. Thus at any given time, the collection of locations from which calls are being made is exactly an independent set in the graph. If calls are attempted independently at each vertex as a Poisson process of rate and have independent exponential mean lengths, it can be shown that the long-run stationary distribution of this process is the hard-core measure on .

Our particular focus in this paper is the mixing time of the Glauber dynamics, or single-site update Markov chain, for this model. The measure can be realized as the stationary distribution of a certain Markov chain. Specifically, consider the chain on state space with transition probabilities given by

Underpinning the definition of is the following dynamical process, known as the Glauber dynamics on . From an independent set , the process follows three steps. The first step is to choose a vertex uniformly from . The second step is to “add” to with probability , and “remove” it with probability ; that is, to set

The third step is to move to if it is a valid independent set, and stay at otherwise.

It is readily checked that is an ergodic, aperiodic, time reversible Markov chain with (unique) stationary distribution . A natural question to ask about is how quickly it converges to its stationary distribution. To make this question precise, we need a few definitions.

Let be an ergodic Markov chain on state space , with transition probabilities . For a state , denote by the distribution of the state at time , given that the initial state is , and denote by the stationary distribution. Define the mixing time of by

The mixing time of captures the speed at which the chain converges to its stationary distribution: for every , in order to get a sample from which is within of (in variation distance), it is necessary and sufficient to run the chain from some arbitrarily chosen distribution for some multiple (depending on ) of the mixing time.

Much work has been done on the question of bounding . The strongest general result available to date is due to Vigoda [21] who showed that if is any -vertex graph with maximum degree , then whenever . In the other direction, Dyer, Frieze and Jerrum [9] considered the case and showed that for each a random (uniform) -regular, -vertex bipartite almost surely (with probability tending to as tends to infinity) satisfies for some absolute constant .

Here, we continue in the spirit of [9] and construct explicit families of graphs for which Glauber dynamics mixes slowly. Specifically, we establish a certain expansion condition in a regular bipartite graph that forces to be (almost) exponential in provided is suitably large (as a function of the expansion). The -dimensional hypercube satisfies this condition for .

Our work is partly motivated by [5] where a study was made of Glauber dynamics for the hard-core measure on the even discrete torus . This is the graph on (with even) in which two strings are adjacent if they differ on only one coordinate, and differ by on that coordinate. It was shown in [5] that for growing exponentially in (with a suitably large base), is exponential in for some that depends on but not on .

In light of a recent result of Galvin and Kahn [12], we found it tempting to believe that slow mixing on should hold for much smaller values of ; even for some values of tending to as grows. The main result of [12] is that the hard-core model on exhibits multiple Gibbs phases for for some large constant . Specifically, write and for the sets of even and odd vertices of (defined in the obvious way: a vertex of is even if the sum of its coordinates is even). Set

For , choose from with . The main result of [12] is that there is a constant such that if then

Thus, roughly speaking, the influence of the boundary on behavior at the origin persists as the boundary recedes. Informally, this suggests that for in this range, the typical independent set chosen according to the hard-core measure is either predominantly odd or predominantly even. Thus there is a highly unlikely “bottleneck” set of balanced independent sets separating the predominantly odd sets from the predominantly even ones. It is the existence of this bottleneck that should cause the conductance of the Glauber dynamics chain to be small (see Section 2), and thus cause its mixing time to be large.

Our main result (Theorem 1.1) provides some support for this belief, verifying it in the case ; unfortunately, because of the weak isoperimetry of the torus we cannot hope to use Theorem 1.1 to deal with general . (See Remark 1.6 for further discussion of these issues.)

Before stating Theorem 1.1, we establish some notation. From now on, will be a -regular, bipartite graph with partition classes and . Set and .

For we write if there is an edge in joining and . Set ( is the neighbourhood of ) and . For (or ) set

we think of as an “external closure” of . Note that while determines , determines only . For this reason, we find it more convenient at some points in the sequel to deal with rather than with itself. Say that is small if . Define the bipartite expansion constant of by

Note that . (The second inequality is obvious. The first follows from regularity, which implies that has a perfect matching, which in turn implies that for all (or ), .)

All implied constants in and notation are independent of . We use “” throughout for and “” for . We write for . We always assume that is sufficiently large to support our assertions.

Set

(2) |

and

We note for future reference that

(3) |

Our main result is

###### Theorem 1.1

Let be a -regular, bipartite graph with vertices and bipartite expansion constant . There is a constant such that whenever and satisfy

(4) |

we have

###### Remark 1.2

If we add as an additional hypothesis to Theorem 1.1 that has bounded codegree (that is, there is a constant independent of such that each pair of vertices in has at most common neighbours), then we can slightly improve our bound on to

(5) |

We do not present the more complicated argument here.

Note that since , we cannot possibly satisfy (4) for , so we may (and will) assume from here on that .

A slightly stronger condition that implies (4) is

(6) |

where the constant depends on , from which we can more clearly see the tradeoff between and . From (6) we may also read off the following corollary of Theorem 1.1 addressing Glauber dynamics for sampling a uniform independent set ().

###### Corollary 1.3

Let satisfy the conditions of Theorem 1.1. There is a constant such that whenever we have

As an application of Theorem 1.1, we consider the case , the -dimensional hypercube. This is the -regular bipartite graph on vertex set in which two vertices are adjacent if they differ on exactly one coordinate. The hypercube satisfies (see, e.g. [16, Lemma 1.3]; this bound can also be derived from isoperimetric inequalities of Bezrukov [2] and Körner and Wei [15]) and so if is a suitably large constant (depending on the constant provided by Theorem 1.1) then (4) is satisfied as long as . So the following is a corollary of Theorem 1.1.

###### Corollary 1.4

There are constants such that whenever we have

In particular,

###### Remark 1.5

###### Remark 1.6

Let us return to , the even discrete torus. Since is easily seen to be isomorphic to , Corollary 1.4 gives an exponential lower bound on for sufficiently large whenever . Unfortunately, the best bound we can obtain on the bipartite expansion constant of is (see, e.g [12]), so we cannot use Theorem 1.1 to obtain any lower bound on independent of beyond which is large for all even and sufficiently large . However, subsequent to the completion of this paper, a strategy specific to the torus has been employed in [11] to show that for all even , and sufficiently large,

## 2 Proof of Theorem 1.1

The notion of conductance, introduced in [13], can be used to analyze the behavior of . Let be a Markov chain on state space with transition matrix and stationary distribution . For and , set

For , define the conductance of as

We may interpret as the probability under that the chain escapes from in one step, given that it is in . Define the conductance of as

We may then bound the mixing time of by

(7) |

(see e.g. [9], where the above bound is derived without assuming time-reversibility of the chain ). Thus to show that the mixing time is large, it is enough to exhibit a single with small conductance.

Throughout this section we fix satisfying the conditions of Theorem 1.1. Set

define analogously, and set ( is the set of balanced independent sets). Without loss of generality, assume . Because Glauber dynamics changes the size of an independent set by at most one at each step, we have that if satisfy , then . It follows that

The simplest way to see (2) is to use the fact that is time-reversible (that is, that for all ); but note that more generally if is a (not necessarily time-reversible) Markov chain on finite state space with transition matrix and stationary distribution then

for all . Now using the trivial lower bound (recall that for , ) we obtain

(9) |

Thus (recalling (7)) to show that is large, it is enough to show that is small. We may think of as a “bottleneck” set through which any run of the chain must pass in order to mix; if the bottleneck has low measure, the mixing time is high.

We will actually consider a larger “bottleneck” set. Set

and

where is as defined in (2). Note that . We will show that as long as satisfies (4),

(10) |

Dealing with is relatively straightforward. We begin by observing that

(11) |

To see this, first set

Note that for all , . We have and so (11) is equivalent to

which is in turn equivalent to

That this inequality holds for all is a routine calculus exercise. Note also that for ,

(12) |

(where recall is the usual binary entropy function). Finally, we use a result concerning the sums of binomial coefficients which follows from the Chernoff bounds [6] (see also [4], p.11):

(13) |

where denotes the integer part of .

Now with the inequalities justified below, we have

(14) | |||||

(15) | |||||

(16) | |||||

(17) | |||||

(18) |

Here (and throughout) we use for . In (14), we are using (13), which is applicable by (3). In (15) we are using the first inequality in (12) and in (16) we are using the second (again, both of these are applicable by (3).) Finally in (17) we are using (11).

Bounding requires much more work. We begin by enlarging slightly. Say that is small on if (recall that for , ), and set

Define small on and similarly. A simple argument, based on the fact that has a perfect matching, shows that any must be small on at least one of , , and so we have

We may assume without loss of generality that

so that it is enough to show that

For each and set

and set

We have

The key now is to upper bound . The following theorem (whose proof is given in Section 3) is based on ideas of A. Sapozhenko [18, 19].

###### Theorem 2.1

## 3 Proof of Theorem 2.1

For and we write for the set of edges having one end in and (if ) for the set of edges having one end in each of . We also write for .

Throughout this section, we fix satisfying the assumptions of Theorem 1.1. We also fix and , but we do not assume . We write for . Given we always write for and set . Note that for ,

(20) |

The proof of Theorem 2.1 involves the idea of “approximation”. We begin with an informal outline. To bound , we produce a small set with the properties that each is “approximated” (in an appropriate sense) by some , and for each , the total weight of those that could possibly be “approximated” by is small. (Each will consist of two parts; one each approximating and .) The product of the bound on and the bound on the weight of those that may be approximated by any is then a bound on . The set is itself produced by an approximation process — we first produce a small set with the property that each is “weakly approximated” (in an appropriate sense) by some , and then show that for each there is a small set with the property that for each that is “weakly approximated” by , there is a which approximates ; we then take . (Each will consist of a single part.)

The main inspiration for the proof of Theorem 2.1 is the work of A. Sapozhenko, who, in [19], gave a relatively simple derivation for the asymptotics of the number of independent sets in (in the notation of (1), this is the asymptotics of with ), earlier derived in a more involved way in [16]. Our Lemma 3.3 is a modification of a lemma in [18], and our overall approach is similar to [19]. See e.g. [10] for another recent application of these ideas.

We now begin the formal discussion of Theorem 2.1 by introducing the two notions of approximation that we will use, beginning with the weaker notion. A covering approximation for is a set satisfying

The second notion of approximation depends on a parameter , . A -approximation for is a pair satisfying

(21) |

(22) |

and

(23) |

Note that if then , and if then . If we think of as “approximate ” and as “approximate ”, (22) says that if is in “approximate ” then almost all of its neighbours are in “approximate ”, while (23) says that if is not in “approximate ” then almost all of its neighbours are not in “approximate ”.

Before continuing, we note a property of -approximations that will be of use later.

###### Lemma 3.1

If is a -approximation for then

(24) |

Proof: Observe that is bounded above by and below by , giving

and that each contributes at least edges to , a set of size , giving

These two observations together give (24).

There are three parts to the proof of Theorem 2.1. Lemma 3.2 is the first “approximation” step, producing a small family of covering approximations for . Lemma 3.3 is the second “approximation” step, refining the covering approximations to produce a family of -approximations for . Finally, Lemma 3.4 is the “reconstruction” step, bounding the weight of the set of ’s that could possibly be -approximated by a member of . We now state the three relevant lemmas. We will then derive Theorem 2.1 before turning to the proofs of the approximation and reconstruction lemmas.

###### Lemma 3.2

There is a with

such that each has a covering approximation in .

###### Lemma 3.3

For any and there is a with

such that any for which is a covering approximation has a -approximation in .

###### Lemma 3.4

Given and , for each that satisfies (24) we have

(25) |

where the sum is over all those ’s in satisfying and .

Before turning to the proofs of Lemmas 3.2, 3.3 and 3.4, we use them to obtain Theorem 2.1. Throughout, we will use (usually without comment) a simple observation about sums of binomial coefficients: if , we have

(26) | |||||

Take and

Note that for this choice of and we have , and so

The bound in (25) is therefore at most

(Here we have used (26)). For our choice of and this is at most

which in turn is at most

(27) |

The bounds in Lemmata 3.3 and 3.2 are at most

(28) |

respectively. For the latter bound, we are using the assumption of Theorem 2.1 and the fact that , which together imply (via (3)) that

Combining (28) with (27), we get

Noting that always, we find that if satisfies (4) with a suitably large constant , then

and so we get Theorem 2.1.

Proof of Lemma 3.2: We appeal to a special case of a fundamental result due to Lovász [17] and Stein [20]. For a bipartite graph with bipartition , we say that covers if each has a neighbour in .

###### Lemma 3.5

If as above satisfies for each and for each , then is covered by some with

Applying the lemma with the subgraph of induced by , , and , we find that each has a covering approximation of size at most . Taking to be the set of all subsets of of size at most , the lemma follows.

Proof of Lemma 3.3: We describe an algorithm, which we refer to as the degree algorithm, which produces for input