The Height Fluctuations of an Off-Critical Dimer Model on the Square Grid

Abstract

The dimer model on a planar bipartite graph can be viewed as a random surface measure. We study these fluctuations for a dimer model on the square grid with two different classes of weights and provide a condition for their equivalence. In the thermodynamic limit and scaling window, these height fluctuations are shown to be non-Gaussian.

Appendix

In Appendix, we present the function from Lemma 4.5. For y ₁=x ₂−x ₁, y ₂=x ₃−x ₂ and y ₃=x ₄−x ₃, we have

(A.1)

We also have the following lemma which is based on a crude argument.

Lemma A.1

There exists γ ₁,…,γ ₄, so that

(A.2)

Proof

To show that $\int_{\gamma_{1}} \dots\int_{\gamma_{4}}f_{\lambda _{1},\lambda _{2}}(x_{1},x_{2},x_{3},x_{4}) dx_{4} \dots dx_{1}\neq0$ for some choice of intervals γ _i , choose γ ₁=(0,1/2ⁿ), γ ₂=(1−1/2ⁿ,1), γ ₃=(2,2+1/2ⁿ) and γ ₄=(3,3−1/2ⁿ). We claim that $f_{\lambda _{1},\lambda _{2}}(x_{1},x_{2},x_{3},x_{4})$ is positive over this interval and so the integral over γ ₁,…,γ ₄ is non-zero.

We can bound y ₁ and y ₃ in terms of y ₂ where y ₁,y ₂ and y ₃ are defined above. These are given by

(A.3)

and

(A.4)

We can separate (A.1) into positive terms and negative terms, i.e. we can write (A.1) as g ₊(y ₁,y ₂,y ₃)−g ₋(y ₁,y ₂,y ₃) where g ₋ and g ₊ are positive. A lower bound for the positive terms can be achieved by setting y ₁ and y ₃ to y ₂. An upper bound of the absolute value of the negative terms can be achieved by setting y ₁ and y ₃ to y ₂(2ⁿ−2)/(2ⁿ+2). In other words, we have

(A.5)

and

(A.6)

The difference between these two bounds is positive provided we take n=|λ|m for |λ|≥1 and n=m/|λ| for |λ|<1 where m>20 (these bounds are not tight).

Indeed, by setting (2ⁿ−2)/(2ⁿ+2)=1−x, we can take a series expansion with respect to x between the positive and negative terms, which is given by

(A.7)

where the derivative is with respect to x. The x ⁰ term is strictly positive and is bounded below by $e^{- 12 y_{2} |\lambda|}$. This is due to comparing the following positive and negative terms of (A.1) and noting that the difference in each case is bounded below by $e^{- 12 y_{2} |\lambda|}$. These differences are given explicitly by

(A.8)

(A.9)

(A.10)

(A.11)

For |λ|≥1, the coefficient of the x ¹ term in the above series expansion is negative and its absolute value is bounded above by $e^{-|\lambda| y_{2}}$ for y ₂≥1. As x is smaller than 1/2^20γ, then $f_{\lambda_{1}, \lambda_{2}}(x_{1},x_{2},x_{3},x_{4})$ is positive over γ ₁,…,γ ₄ for |λ|≥1. For 0<|λ|<1, the absolute value of the coefficient of x ¹ is bounded above by some constant c>2 for y ₂≥1. As 1/2^20γ is less than $e^{-12 y_{2} \gamma}$, the order x ¹ term is less than the order x ⁰ term. This means $f_{\lambda_{1}, \lambda_{2}}(x_{1},x_{2},x_{3},x_{4})>0$ over γ ₁,…,γ ₄. □