Harmonic Crystal using Random Walk

Question

Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.

Basically, it is a proof of the instability in an harmonic lattice crystal that uses the idea of random walk and the discrete lagrangian, and it is quite self contained. We were able to somewhat reconstruct it, but a firm reference would obviously help. If my terminology is somewhat vague, here is exactly the theorem we're trying to find its proof:

Consider the lattice $\Lambda = [-L\cdots L]^2 \in \mathbb{Z} ^2 $ and a scalar field $X$ on it, i.e. $ \varphi (x) \in \mathbb{R} $. The particles outside $\Lambda $ are tied down, meaning $ \varphi (x) = 0 $, $\forall x \notin \Lambda $.

Energy will be defined by $\nu (X) = \Sigma _{x \sim y} (\varphi (x) \ - \varphi(y))^2$, sum over all neighboring lattice points. The partition function in the regular way:

$$Z = \int\limits_{\mathbb{R} ^ {|\Lambda|} } dX \exp(-\nu (X)) $$

The theorem is as follows:

For $L \to \infty$, we have that $\langle\varphi (0)^2 \rangle = \int\limits_{\mathbb{R} ^ | \Lambda |} dX \exp(-\nu (X)) \varphi (0) ^2 $ diverges like $\log (| \Lambda | )$

This post imported from StackExchange MathOverflow at 2015-02-16 11:49 (UTC), posted by SE-user user42864

Comments

Can you define $\langle\varphi(0)\rangle$?

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user Liviu Nicolaescu

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Adding it to the original question

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user user42864

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I think you might want $x\sim y$ also for $x-y=(\pm1,\pm1)$. Otherwise $\langle\phi(0)^2\rangle$ will diverge for more boring reasons.

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user Yoav Kallus

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I wrote that this is only for neighbouring $x,y$

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user user42864

Answer 1

The object you look at is called the Gaussian Free Field (on your graph, with zero boundary conditions) in dimension $2$. There is a lot known about it. For some pointers see the Wiki page http://en.wikipedia.org/wiki/Gaussian_free_field, my lecture notes http://www.wisdom.weizmann.ac.il/~zeitouni/notesGauss.pdf and Sznitman's lecture notes https://www.math.ethz.ch/u/sznitman/SpecialTopics.pdf. Your specific question really asks about the Green function for the Laplacian in the two dimensional box, for which detailed results are available in the probabilistic literature on random walks, see Spitzer's book or Lawler's book.

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user ofer zeitouni

Comments

Which books, to be precise?

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user user42864

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Spitzer: principles of random walks (I believe this is the only book Spitzer wrote). Lawler: intersection of random walks

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user ofer zeitouni

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thank you very much!

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user user42864

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Sections 1.5 and 1.6 of the book by Lawler have what you are looking for.

This post imported from StackExchange MathOverflow at 2015-02-16 11:50 (UTC), posted by SE-user Abdelmalek Abdesselam