Conway's 'Angel Problem' and Critical Phenomena?

Hi,

This is a query on possible connections between critical phenomena and
the 'Conway's Angel problem' from game theory. The following two pages
introduce the problem:

mathworld.wolfram.com/AngelProblem.html
www.msri.org/publications/books/Book29/files/conway.pdf

In brief, the angel and devil play a game on a 2d infinite chessboard
and the angel tries to escape and the devil tries to trap him in some
finite region. On a 1d lattice the devil wins trivially. In 3d, the
angel has plenty of space to move (it seems). but 2d remains open. Full
details are in above sources, especially the latter.

My doubt: Are there possible connections between some statistical
mechanics model (possibly possessing criticality) and the Angel
problem?

As far as i know, one strong (if not defining) feature of criticality
is the absence of a length scale in the system and it appearing the
same at all length scales - in other words it is a fractal.

Conway's paper mentioned above describes the angel's path in a 2d board
which can give him *some chance* of escape. His description of the
angel's path (sections 7 and 9 of the paper) appears to show fractal
property. so, one feels the system is critical - there presumably is
some mapping between some statistical model that exhibits criticality
and the game. Moreover, there is a general strategy for the Angel ,
attributed to Korner which seems to sweep over all length scales of the
system - reminiscent of Scaling theory (?).

There is the issue of interpreting the criticality (assuming it exists
in 2d). The guess is that the scenario is somewhat like this:

- If the game were played on a 1d lattice, whatever the angel does, the
devil traps him with infinitely many moves to spare.
- In a 3d lattice of cubes , the angel can win and at every stage he
has many cubes to move into and his path can fill a 3d subspace of the
full space.
- In 2d chessboard , the situation is perhaps critical - the angel can
avoid getting trapped but gets confined to a 1d subspace of the 2d
space; he needs to stay on and move on a 1-d fractal and any deviation
from this path traps him.

I feel phenomena such as 'Percolation' and 'Anderson localization' in
statistical mechanics might have some correspondence to the angel's
freedom of movement.

I request comments/corrections.

With regards,
R. Nandakumar.

.

Relevant Pages

  • Re: God killing Himself to save us - a logical theory?
    ... > You look like an angel ... > You're the devil in disguise ... >> about God killing himself. ...
    (soc.culture.singapore)
  • Re: Deserts and Disharmony C&C please
    ... The "message" can be cobbled around until it matches the observed person, or until it maintains its question, and the Devil can thus address either in absentia. ... A Master can maintain himself and one chancery Angel; a Grandmaster can maintain himself and two, arguing with both "simultaneously" without dropping pieces or mixing them up; etc. ... the nightmare of their deaths and the creativity to live a new dream. ... The "poet" who provides the best padding at the least personal cost to the Baby is the one whose collection plat... ...
    (rec.arts.poems)
  • Re: God killing Himself to save us - a logical theory?
    ... >> You look like an angel ... >> You're the devil in disguise ... >>> The Demons inspired certain people to oppose God and to kill the Son ...
    (soc.culture.singapore)
  • Re: Penny or Smoltz
    ... We'll also get to see how good Smoltz looks too. ... Devil or angel, I can't make up my mind ... Devil or angel please say you'll be mi-i-ine ...
    (alt.sports.baseball.bos-redsox)
  • Re: Shields
    ... Joe could have ... He acted like it was a mid season game against Baltimore. ... > Angel fan here. ... I have confidence in the Yanks ...
    (alt.sports.baseball.ny-yankees)