From: marty@amaterasu.physics.uiuc.edu (Marty Gelfand)
Subject: Re: a first year grad student freaks out.
Organization: Department of Physics, University of Illinois at Urbana-Champaign
Date: Thu, 12 Nov 1992 02:33:26 GMT

In article <BxKu9s.BAw@news.udel.edu> Ray J Cornwall writes:
>John C. Baez writes:
>>That's why I'm glad that starting in high school I started trying to
>>come up with math topics to do research on.  One has to face the unknown
>>eventually if one is going to do research, so one might as well get used
>>to it as early as possible (even if ones first research projects are a
>>little silly).
>
>For those of us who wish to do research (cause we'll get our names in
>really cool journals! :) can you give us some pointers on how to come
>up with such topics?
>I've been trying to be more aggressive with my studies in math, to
>look for such ideas, but I have a feeling I'm barking up the wrong
>tree looking in textbooks and looking for my favorite topics here.
>Where are some good sources of information, and what are some good
>questions to ask myself?
  Here's an idea JCB would probably appreciate...talk to a theoretical
physicist every once in a while.  We're full of problems we can't solve
and solutions we can't rigorously justify.  (Has anyone put the Parisi
solution of the Sherrington-Kirkpatrick [infinite-dimensional Ising spin
glass] model on a mathematically sound footing yet?)  To give you some
idea how much territory is wide open, here's a problem I worried about
two years ago (for the physical motivation and some Monte Carlo work,
see Zhao et al, Phys Rev B 44, 10760, 1991).  Please let me know if
you make any substantial progress!  I suspect it will not be easy.
  Consider the set of directed random walks on the lattice of integer
points in two dimensions from (0,0) to (t,t).  Suppose that each lattice
point is associated with a number v(i,j), where v=1 with probability 1-p
and -1 with probability p. (Actually, let me amend that, and say that
v(0,0)=1 always.  Also let's assume that p is in the interval [0,1/2].)  
  Now each directed walk W can be associated with a 
number V(W), which is the product of all the v's encountered along the walk.
Now add up all the [2t choose t] V(W)'s, which defines the 'partition
function' Z (which depends on the realization of the v(i,j)'s and on t).
Define the 'sign probability' S(p) as the t->infinity limit of the
expectation value of sgn(Z).
  Conjecture: For any p>0, S(p)=0.
  Interesting generalizations: change the dimensionality of the lattice.
For the 1D case the Conjecture is obviously true.  For d>=4 there is
reason to suspect that a mean-field argument, which yields a p_c>0 such
that S(p)=0 for p<=p_c and S(p)>0 for p>p_c, is valid (but it isn't
what mathematicians would regard as a rigourous proof).  What about d=3?
  Another extension:  Consider not only the large-t limit, but also the
asymptotic behavior, ie, define S(p;t) as the expectation value of
sgn(Z(t)), and determine the large-t asymptotics.  Based on numerical
work and some theoretical prejudices I expect that in d=2
S(p;t) ~ exp(t/L)/t^M, where L depends strongly on p as p->0 (probably 
diverging like exp(1/p)) and M is independent of p.  Of course this
guess could be wrong.

  --Marty Gelfand   marty@amaterasu.physics.uiuc.edu