On the sampling problem for 𝐻-colorings on the hypercubic lattice

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

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Evaluation of the Watson integral and associated logarithmic integral for thed-dimensional hypercubic lattice

Journal of Physics A General Physics ◽

10.1088/0305-4470/34/36/314 ◽

2001 ◽

Vol 34 (36) ◽

pp. 7349-7354 ◽

Cited By ~ 24

Author(s):

G S Joyce ◽

I J Zucker

Keyword(s):

Hypercubic Lattice ◽

Logarithmic Integral

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Linear sampling and the ∀∃∀ case of the decision problem

Journal of Symbolic Logic ◽

10.2307/2272894 ◽

1974 ◽

Vol 39 (3) ◽

pp. 519-548 ◽

Cited By ~ 10

Author(s):

Stål O. Aanderaa ◽

Harry R. Lewis

Keyword(s):

Decision Problem ◽

Combinatorial Problem ◽

Finite Model ◽

Unsolvable Problem ◽

Sampling Problem ◽

Solvable Class ◽

Full Class ◽

Infinite Model ◽

Intricate Argument ◽

Linear Sampling

Let Q be the class of closed quantificational formulas ∀x∃u∀yM without identity such that M is a quantifier-free matrix containing only monadic and dyadic predicate letters and containing no atomic subformula of the form Pyx or Puy for any predicate letter P. In [DKW] Dreben, Kahr, and Wang conjectured that Q is a solvable class for satisfiability and indeed contains no formula having only infinite models. As evidence for this conjecture they noted the solvability of the subclass of Q consisting of those formulas whose atomic subformulas are of only the two forms Pxy, Pyu and the fact that each such formula that has a model has a finite model. Furthermore, it seemed likely that the techniques used to show this subclass solvable could be extended to show the solvability of the full class Q, while the syntax of Q is so restricted that it seemed impossible to express in formulas of Q any unsolvable problem known at that time.In 1966 Aanderaa refuted this conjecture. He first constructed a very complex formula in Q having an infinite model but no finite model, and then, by an extremely intricate argument, showed that Q (in fact, the subclass Q2 defined below) is unsolvable ([Aa1], [Aa2]). In this paper we develop stronger tools in order to simplify and extend the results of [Aa2]. Specifically, we show the unsolvability of an apparently new combinatorial problem, which we shall call the linear sampling problem (defined in §1.2 and §2.3). From the unsolvability of this problem there follows the unsolvability of two proper subclasses of Q, which we now define. For each i ≥ 0, let Pi be a dyadic predicate letter and let Ri be a monadic predicate letter.

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SU(2) quark potential on a body-centered-hypercubic lattice

Physical Review D ◽

10.1103/physrevd.33.3718 ◽

1986 ◽

Vol 33 (12) ◽

pp. 3718-3732 ◽

Cited By ~ 1

Author(s):

William Celmaster ◽

K. J. M. Moriarty

Keyword(s):

Hypercubic Lattice ◽

Quark Potential

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Off the Record: Unrecorded Legislative Votes, Selection Bias and Roll-Call Vote Analysis

British Journal of Political Science ◽

10.1017/s0007123406000366 ◽

2006 ◽

Vol 36 (4) ◽

pp. 691-704 ◽

Cited By ~ 155

Author(s):

CLIFFORD J. CARRUBBA ◽

MATTHEW GABEL ◽

LACEY MURRAH ◽

RYAN CLOUGH ◽

ELIZABETH MONTGOMERY ◽

...

Keyword(s):

Random Sample ◽

Selection Bias ◽

European Parliament ◽

Roll Call ◽

General Study ◽

Roll Call Vote ◽

Legislative Voting ◽

Sampling Problem ◽

The Universe

Scholars often use roll-call votes to study legislative behaviour. However, many legislatures only conclude a minority of decisions by roll call. Thus, if these votes are not a random sample of the universe of votes cast, scholars may be drawing misleading inferences. In fact, theories over why roll-call votes are requested would predict selection bias based on exactly the characteristics of legislative voting that scholars have most heavily studied. This article demonstrates the character and severity of this sampling problem empirically by examining European Parliament vote data for a whole year. Given that many legislatures decided only a fraction of their legislation by roll call, these findings have potentially important implications for the general study of legislative behaviour.

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