arXiv subjects

David Galvin

Publications and source records attributed to David Galvin.

52 records · Page 3Linked to original sources

arXiv2012

On homomorphisms from the Hamming cube to {\bf Z}

Write ${\cal F}$ for the set of homomorphisms from $\{0,1\}^d$ to ${\bf Z}$ which send $\underline{0}$ to 0 (think of members of ${\cal F}$ as labellings of $\{0,1\}^d$ in which adjacent strings get labels differing by exactly 1), and ${\cal F}_i$ for those which take on exactly $i$ values. We give asymptotic formulae for $|{\cal F}|$ and $|{\cal F}_i|$. In particular, we show that the probability that a uniformly chosen member ${\bf f}$ of ${\cal F}$ takes more than five values tends to 0 as $d \rightarrow \infty$. This settles a conjecture of J. Kahn. Previously, Kahn had shown that there is a constant $b$ such that ${\bf f}$ a.s. takes at most $b$ values. This in turn verified a conjecture of I. Benjamini {\em et al.}, that for each $t > 0$, ${\bf f}$ a.s. takes at most $td$ values. Determining $|{\cal F}|$ is equivalent both to counting the number of rank functions on the Boolean lattice $2^{[d]}$ (functions $f \colon 2^{[d]} \longrightarrow {\bf N}$ satisfying $f(\emptyset)=0$ and $f(A) \leq f(A \cup x) \leq f(A)+1$ for all $A \in 2^{[d]}$ and $x \in [d]$) and to counting the number of proper 3-colourings of the discrete cube (i.e., the number of homomorphisms from $\{0,1\}^d$ to $K_3$, the complete graph on 3 vertices). Our proof uses the main lemma from Kahn's proof of constant range, together with some combinatorial approximation techniques introduced by A. Sapozhenko.

math.CO↗

arXiv2012

On weighted graph homomorphisms

For given graphs $G$ and $H$, let $|Hom(G,H)|$ denote the set of graph homomorphisms from $G$ to $H$. We show that for any finite, $n$-regular, bipartite graph $G$ and any finite graph $H$ (perhaps with loops), $|Hom(G,H)|$ is maximum when $G$ is a disjoint union of $K_{n,n}$'s. This generalizes a result of J. Kahn on the number of independent sets in a regular bipartite graph. We also give the asymptotics of the logarithm of $|Hom(G,H)|$ in terms of a simply expressed parameter of $H$. We also consider weighted versions of these results which may be viewed as statements about the partition functions of certain models of physical systems with hard constraints.

math.CO↗

arXiv2012

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

Let $\gS=(V,E)$ be a finite, $d$-regular bipartite graph. For any $λ>0$ let $π_λ$ be the probability measure on the independent sets of $\gS$ in which the set $I$ is chosen with probability proportional to $λ^{|I|}$ ($π_λ$ is the {\em hard-core measure with activity $λ$ on $\gS$}). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is $π_λ$. We show that when $λ$ is large enough (as a function of $d$ and the expansion of subsets of single-parity of $V$) then the convergence to stationarity is exponentially slow in $|V(\gS)|$. In particular, if $\gS$ is the $d$-dimensional hypercube $\{0,1\}^d$ we show that for values of $λ$ tending to 0 as $d$ grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods.

math.CO↗

arXiv2012

Torpid Mixing of Local Markov Chains on 3-Colorings of the Discrete Torus

We study local Markov chains for sampling 3-colorings of the discrete torus $T_{L,d}={0,..., L-1}^d$. We show that there is a constant $ρ\approx .22$ such that for all even $L \geq 4$ and $d$ sufficiently large, certain local Markov chains require exponential time to converge to equilibrium. More precisely, if $\cM$ is a Markov chain on the set of proper 3-colorings of $T_{L,d}$ that updates the color of at most $ρL^d$ vertices at each step and whose stationary distribution is uniform, then the convergence to stationarity of $\cM$ is exponential in $L^{d-1}$. Our proof is based on a conductance argument that builds on sensitive new combinatorial enumeration techniques.

math.CO↗

arXiv2012

Bounding the partition function of spin-systems

With a graph $G=(V,E)$ we associate a collection of non-negative real weights $\cup_{v\in V}{λ_{i,v}:1\leq i \leq m} \cup \cup_{uv \in E} {λ_{ij,uv}:1\leq i \leq j \leq m}$. We consider the probability distribution on ${f:V\rightarrow{1,...,m}}$ in which each $f$ occurs with probability proportional to $\prod_{v \in V}λ_{f(v),v}\prod_{uv \in E}λ_{f(u)f(v),uv}$. Many well-known statistical physics models, including the Ising model with an external field and the hard-core model with non-uniform activities, can be framed as such a distribution. We obtain an upper bound, independent of $G$, for the partition function (the normalizing constant which turns the assignment of weights on $\{f:V\rightarrow{1,...,m\}}$ into a probability distribution) in the case when $G$ is a regular bipartite graph. This generalizes a bound obtained by Galvin and Tetali who considered the simpler weight collection $\{λ_i:1 \leq i \leq m\} \cup \{λ_{ij}:1 \leq i \leq j \leq m\}$ with each $λ_{ij}$ either 0 or 1 and with each $f$ chosen with probability proportional to $\prod_{v \in V}λ_{f(v)}\prod_{uv \in E}λ_{f(u)f(v)}$. Our main tools are a generalization to list homomorphisms of a result of Galvin and Tetali on graph homomorphisms and a straightforward second-moment computation.

math.CO↗

arXiv2012

Sampling 3-colourings of regular bipartite graphs

We show that if $\gS=(V,E)$ is a regular bipartite graph for which the expansion of subsets of a single parity of $V$ is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise non-adjacent vertices), and if $\cM$ is a Markov chain on the set of proper 3-colourings of $\gS$ which updates the colour of at most $ρ|V|$ vertices at each step and whose stationary distribution is uniform, then for $ρ\approx .22$ and $d$ sufficiently large the convergence to stationarity of $\cM$ is (essentially) exponential in $|V|$. In particular, if $\gS$ is the $d$-dimensional hypercube $Q_d$ (the graph on vertex set $\{0,1\}^d$ in which two strings are adjacent if they differ on exactly one coordinate) then the convergence to stationarity of the well-known Glauber (single-site update) dynamics is exponentially slow in $2^d/(\sqrt{d}\log d)$. A combinatorial corollary of our main result is that in a uniform 3-colouring of $Q_d$ there is an exponentially small probability (in $2^d$) that there is a colour $i$ such the proportion of vertices of the even subcube coloured $i$ differs from the proportion of the odd subcube coloured $i$ by at most $.22$. Our proof combines a conductance argument with combinatorial enumeration methods.

math.CO↗

arXiv2012

Two problems on independent sets in graphs

Let $i_t(G)$ denote the number of independent sets of size $t$ in a graph $G$. Levit and Mandrescu have conjectured that for all bipartite $G$ the sequence $(i_t(G))_{t \geq 0}$ (the {\em independent set sequence} of $G$) is unimodal. We provide evidence for this conjecture by showing that is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph $G(n,n,p)$, and show that for any fixed $p\in(0,1]$ its independent set sequence is almost surely unimodal, and moreover almost surely log-concave except perhaps for a vanishingly small initial segment of the sequence. We obtain similar results for $p=\tildeΩ(n^{-1/2})$. We also consider the problem of estimating $i(G)=\sum_{t \geq 0} i_t(G)$ for $G$ in various families. We give a sharp upper bound on the number of independent sets in an $n$-vertex graph with minimum degree $δ$, for all fixed $δ$ and sufficiently large $n$. Specifically, we show that the maximum is achieved uniquely by $K_{δ, n-δ}$, the complete bipartite graph with $δ$ vertices in one partition class and $n-δ$ in the other. We also present a weighted generalization: for all fixed $x>0$ and $δ>0$, as long as $n=n(x,δ)$ is large enough, if $G$ is a graph on $n$ vertices with minimum degree $δ$ then $\sum_{t \geq 0} i_t(G)x^t \leq \sum_{t \geq 0} i_t(K_{δ, n-δ})x^t$ with equality if and only if $G=K_{δ, n-δ}$.

math.CO↗

arXiv2012

Matchings and Independent Sets of a Fixed Size in Regular Graphs

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size $\ell$ in the graph consisting of $\frac{N}{2d}$ disjoint copies of the complete bipartite graph $K_{d,d}$. This provides asymptotic evidence for a conjecture of S. Friedland {\it et al.}. We also obtain an analogous result for independent sets of a fixed size in regular graphs, giving asymptotic evidence for a conjecture of J. Kahn. Our bounds on the number of matchings and independent sets of a fixed size are derived from bounds on the partition function (or generating polynomial) for matchings and independent sets.

math.CO↗

arXiv2012

Counting colorings of a regular graph

At most how many (proper) q-colorings does a regular graph admit? Galvin and Tetali conjectured that among all n-vertex, d-regular graphs with 2d|n, none admits more q-colorings than the disjoint union of n/2d copies of the complete bipartite graph K_{d,d}. In this note we give asymptotic evidence for this conjecture, giving an upper bound on the number of proper q-colorings admitted by an n-vertex, d-regular graph of the form a^n b^{n(1+o(1))/d} (where a and b depend on q and where o(1) goes to 0 as d goes to infinity) that agrees up to the o(1) term with the count of q-colorings of n/2d copies of K_{d,d}. An auxiliary result is an upper bound on the number of colorings of a regular graph in terms of its independence number. For example, we show that for all even q and fixed ε> 0 there is δ=δ(ε,q) such that the number of proper q-colorings admitted by an n-vertex, d-regular graph with no independent set of size n(1-ε)/2 is at most (a-δ)^n.

math.CO↗

arXiv2012

Counting independent sets of a fixed size in graphs with a given minimum degree

Galvin showed that for all fixed $δ$ and sufficiently large $n$, the $n$-vertex graph with minimum degree $δ$ that admits the most independent sets is the complete bipartite graph $K_{δ,n-δ}$. He conjectured that except perhaps for some small values of $t$, the same graph yields the maximum count of independent sets of size $t$ for each possible $t$. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples $(n,δ, t)$ with $t\geq 3$, no $n$-vertex {\em bipartite} graph with minimum degree $δ$ admits more independent sets of size $t$ than $K_{δ,n-δ}$. Here we make further progress. We show that for all triples $(n,δ,t)$ with $δ\leq 3$ and $t\geq 3$, no $n$-vertex graph with minimum degree $δ$ admits more independent sets of size $t$ than $K_{δ,n-δ}$, and we obtain the same conclusion for $δ> 3$ and $t \geq 2δ+1$. Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree $δ$ whose minimum degree drops on deletion of an edge or a vertex.

math.CO↗

arXiv2011

Reverse mathematics and infinite traceable graphs

This paper falls within the general program of investigating the proof theoretic strength (in terms of reverse mathematics) of combinatorial principals which follow from versions of Ramsey's theorem. We examine two statements in graph theory and one statement in lattice theory proved by Galvin, Rival and Sands \cite{GRS:82} using Ramsey's theorem for 4-tuples. Our main results are that the statements concerning graph theory are equivalent to Ramsey's theorem for 4-tuples over $\RCA$ while the statement concerning lattices is provable in $\RCA$. Revised 12/2010. To appear in Archive for Mathematical Logic

math.LO↗

arXiv2010

The multi-state hard core model on a regular tree

The classical hard core model from statistical physics, with activity $λ> 0$ and capacity $C=1$, on a graph $G$, concerns a probability measure on the set ${\mathcal I}(G)$ of independent sets of $G$, with the measure of each independent set $I \in {\mathcal I}(G)$ being proportional to $λ^{|I|}$. Ramanan et al. proposed a generalization of the hard core model as an idealized model of multicasting in communication networks. In this generalization, the {\em multi-state} hard core model, the capacity $C$ is allowed to be a positive integer, and a configuration in the model is an assignment of states from $\{0,\ldots,C\}$ to $V(G)$ (the set of nodes of $G$) subject to the constraint that the states of adjacent nodes may not sum to more than $C$. The activity associated to state $i$ is $λ^{i}$, so that the probability of a configuration $σ:V(G)\rightarrow \{0,\ldots, C\}$ is proportional to $λ^{\sum_{v \in V(G)} σ(v)}$. In this work, we consider this generalization when $G$ is an infinite rooted $b$-ary tree and prove rigorously some of the conjectures made by Ramanan et al. In particular, we show that the $C=2$ model exhibits a (first-order) phase transition at a larger value of $λ$ than the $C=1$ model exhibits its (second-order) phase transition. In addition, for large $b$ we identify a short interval of values for $λ$ above which the model exhibits phase co-existence and below which there is phase uniqueness. For odd $C$, this transition occurs in the region of $λ= (e/b)^{1/\ceil{C/2}}$, while for even $C$, it occurs around $λ=(\log b/b(C+2))^{2/(C+2)}$. In the latter case, the transition is first-order.

math.PR↗

arXiv2010

An upper bound for the number of independent sets in regular graphs

Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If true, this bound would be tight, being achieved by the disjoint union of $N/2d$ copies of $K_{d,d}$. Kahn established the bound for bipartite $G$, and later gave an argument that established $$ i(G)\leq 2^{\frac{N}{2}\left(1+\frac{2}{d}\right)} $$ for $G$ not necessarily bipartite. In this note, we improve this to $$ i(G)\leq 2^{\frac{N}{2}\left(1+\frac{1+o(1)}{d}\right)} $$ where $o(1) \rightarrow 0$ as $d \rightarrow \infty$, which matches the conjectured upper bound in the first two terms of the exponent. We obtain this bound as a corollary of a new upper bound on the independent set polynomial $P(λ,G)=\sum_{I \in {\cal I}(G)} λ^{|I|}$ of an $N$-vertex, $d$-regular graph $G$, namely $$ P(\gl,G) \leq (1+\gl)^{\frac{N}{2}} 2^{\frac{N(1+o(1))}{2d}} $$ valid for all $\gl > 0$. This also allows us to improve the bounds obtained recently by Carroll, Galvin and Tetali on the number of independent sets of a fixed size in a regular graph.

math.CO↗

arXiv2010

Sampling independent sets in the discrete torus

The even discrete torus is the graph T_{L,d} on vertex set {0,...,L-1}^d (L even) with two vertices adjacent if they differ by 1 (mod L) on one coordinate. The hard-core measure with activity x on T_{L,d} is the distribution pi_x on the independent sets (sets of vertices spanning no edges) of T_{L,d} in which a set I is chosen with probability proportional to x^|I|. This distribution occurs in problems from statistical physics and communication networks. We study Glauber dynamics, a single-site update Markov chain on the set of independent sets of T_{L,d} whose stationary distribution is pi_x. We show that for x > cd^{-1/4}log^{3/4}d (and d large) the convergence to stationarity is exponentially slow in L^{d-1}. This improves a result of Borgs et al., who had shown slow mixing for x > c^d. Our proof, which extends to r-local chains (chains which alter the state of at most a proportion r of the vertices in each step) for suitable r, follows the conductance argument of Borgs et al., adding to it some combinatorial enumeration methods that are modifications of those used by Galvin and Kahn to show that the hard-core model with parameter x on the integer lattice Z^d exhibits phase coexistence for x > cd^{-1/4}log^{3/4}d. The graph T_{L,d} is bipartite, with partition classes E (the vertices the sum of whose coordinates is even) and O. Our result can be expressed combinatorially as the statement that for each sufficiently large x, there is an r(x)>0 such that if I is an independent set chosen according to pi_x, then the probability that ||I \cap E|-|I \cap O|| is at most r(x)L^d is exponentially small in L^{d-1}. In particular, for all eps>0 the probability that a uniformly chosen independent set from T_{L,d} satisfies ||I \cap E|-|I \cap O|| \leq (.25 - eps)L^d is exponentially small in L^{d-1}.

math.CO↗

arXiv2010

The number of independent sets in a graph with small maximum degree

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $$ {\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} $$ (where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices in one partition class and $b$ in the other), with equality if and only if each connected component of $G$ is either a complete bipartite graph or a single vertex. This bound (for all $G$) was conjectured by Kahn. A corollary of our result is that if $G$ is $d$-regular with $1 \leq d \leq 5$ then $$ {\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d}, $$ with equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of $K_{d,d}$. This bound (for all $d$) was conjectured by Alon and Kahn and recently proved for all $d$ by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most $3$ the search could be done by hand, but for the case of maximum degree $4$ or $5$, a computer is needed.

math.CO↗

arXiv2010

A threshold phenomenon for random independent sets in the discrete hypercube

Let $I$ be an independent set drawn from the discrete $d$-dimensional hypercube $Q_d=\{0,1\}^d$ according to the hard-core distribution with parameter $λ>0$ (that is, the distribution in which each independent set $I$ is chosen with probability proportional to $λ^{|I|}$). We show a sharp transition around $λ=1$ in the appearance of $I$: for $λ>1$, $\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}=0$ asymptotically almost surely, where ${\cal E}$ and ${\cal O}$ are the bipartition classes of $Q_d$, whereas for $λ<1$, $\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}$ is asymptotically almost surely exponential in $d$. The transition occurs in an interval whose length is of order $1/d$. A key step in the proof is an estimation of $Z_λ(Q_d)$, the sum over independent sets in $Q_d$ with each set $I$ given weight $λ^{|I|}$ (a.k.a. the hard-core partition function). We obtain the asymptotics of $Z_λ(Q_d)$ for $λ>\sqrt{2}-1$, and nearly matching upper and lower bounds for $λ\leq \sqrt{2}-1$, extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution. We also derive a long-range influence result. For all fixed $λ>0$, if $I$ is chosen from the independent sets of $Q_d$ according to the hard-core distribution with parameter $λ$, conditioned on a particular $v \in {\cal E}$ being in $I$, then the probability that another vertex $w$ is in $I$ is $o(1)$ for $w \in {\cal O}$ but $Ω(1)$ for $w \in {\cal E}$.

math.CO↗