Revision #1 Authors: Martin Dyer, Leslie Ann Goldberg, Mark Jerrum

Accepted on: 21st July 2006 00:00

Downloads: 1621

Keywords:

We consider Glauber dynamics on finite spin systems.

The mixing time of Glauber dynamics can be bounded

in terms of the influences of sites on each other.

We consider three parameters bounding these influences ---

$\alpha$, the total influence on a site, as studied by Dobrushin;

$\alpha'$, the total influence of a site, as studied by Dobrushin and Shlosman;

and $\alpha''$, the total influence on a site in any given context,

which is related to the path-coupling method of Bubley and Dyer.

It is known that if any of these parameters is less than~$1$ then

random-update Glauber dynamics (in which a randomly-chosen site is updated

at each step) is rapidly mixing. It is also known that

the Dobrushin condition $\alpha<1$ implies that

systematic-scan Glauber dynamics (in which sites are updated in a deterministic

order) is rapidly mixing.

This paper studies two related issues, primarily in the context of

systematic scan:

(1) the relationship between the parameters~$\alpha$, $\alpha'$ and

$\alpha''$, and

(2) the relationship between proofs of rapid mixing using Dobrushin

uniqueness (which

typically use analysis techniques) and proofs of rapid mixing using path

coupling.

We use matrix-balancing to show that the Dobrushin-Shlosman condition

$\alpha'<1$ implies rapid mixing of systematic scan.

An interesting question is whether the rapid mixing results for scan can be

extended to the $\alpha=1$ or $\alpha'=1$ case.

We give positive results for the rapid mixing of systematic scan for

certain $\alpha=1$

cases.

As an application,

we show rapid mixing of systematic scan (for any scan order) for

heat-bath Glauber dynamics for proper $q$-colourings of a

degree-$\Delta$ graph~$G$

when $q\geq 2\Delta$.

TR05-075 Authors: Martin Dyer, Leslie Ann Goldberg, Mark Jerrum

Publication: 15th July 2005 17:15

Downloads: 1788

Keywords:

We consider Glauber dynamics on finite spin systems.

The mixing time of Glauber dynamics can be bounded

in terms of the influences of sites on each other.

We consider three parameters bounding these influences ---

$\alpha$, the total influence on a site, as studied by Dobrushin;

$\alpha'$, the total influence of a site, as studied by Dobrushin and Shlosman;

and $\alpha''$, the total influence on a site in any given context,

which is related to the path-coupling method of Bubley and Dyer.

It is known that if any of these parameters is less than~$1$ then

random-update Glauber dynamics (in which a randomly-chosen site is updated

at each step) is rapidly mixing. It is also known that

the Dobrushin condition $\alpha<1$ implies that

systematic-scan Glauber dynamics (in which sites are updated in a deterministic

order) is rapidly mixing.

This paper studies two related issues, primarily in the context of systematic scan:

(1) the relationship between the parameters~$\alpha$, $\alpha'$ and $\alpha''$, and

(2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which

typically use analysis techniques) and proofs of rapid mixing using path

coupling.

We use matrix-balancing to show that the Dobrushin-Shlosman condition

$\alpha'<1$ implies rapid mixing of systematic scan.

An interesting question is whether the rapid mixing results for scan can be

extended to the $\alpha=1$ or $\alpha'=1$ case.

We give positive results for the rapid mixing of systematic scan for certain $\alpha=1$

cases.

As an application,

we show rapid mixing of systematic scan (for any scan order) for

heat-bath Glauber dynamics for proper $q$-colourings of a degree-$\Delta$ graph~$G$

when $q\geq 2\Delta$.