On boundedness of zeros of the independence polynomial of tori

Abstract

We study boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. We prove that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori. Here balanced means that the size of the torus is at most exponential in the shortest side length, while highly unbalanced means that the longest side length of the torus is super exponential in the product over the other side lengths cubed. We discuss implications of our results to the existence of efficient algorithms for approximating the independence polynomial on tori. This project was partially inspired by the relationship between zeros of partition functions and holomorphic dynamics, a relationship that in the last two decades played a prominent role in the field. Besides presenting new results, we survey this relationship and its recent consequences.

Figures (6)

• Figure 1: On the left the zeros of partition function of the the $d$-ary trees, and on the right the bifurcation diagram of the functions $f_{\lambda,d}$, for $d=3$. The roots were computed by first exploiting the recursive structure to find an exact formula for the independence polynomial $Z_{T_n}$, and then algebraically solving for the roots of the equation $Z_{T_n} = 0$. The degree and especially the size of the coefficients of these polynomials grow respectively exponentially and super-exponentially with the depth $n$: We have plotted the zeros for depth $n=6$, in which case the independence polynomial has degree $820$, and the coefficients add up to roughly $10^{255}$. The number $n=6$ is clearly far too small to obtain a good idea what happens as $n \rightarrow \infty$. In particular the left hand side plot sheds little light on the known phase transition at $\lambda_+=27/16$. The bifurcation diagram on the right is generated by computing the approximate values of $f^n(\lambda)$ for a discrete raster, and using nearby values to find a discrete approximation of the spherical derivative. Computationally this method is far more efficient. The illustration on the right depicts the bifurcation diagram for $n=200$, and the phase transition at $\lambda_+$ is clearly visible as the right-most point where the bifurcation locus intersects the real axis, which is the horizontal line midway up the figure.
• Figure 2: The figure on the left depicts the zeros of the independence polynomial the subgraph $B_{22}(0)$ in $\mathbb Z^2$: all points in the square lattice whose distance in the graph to the origin is at most $22$. The figure on the right depicts the spherical derivative of the occupation ratio of $(B_{22}(0),0)$.
• Figure 3: The figure on the left depicts the zeros of the independence polynomial of the $2$-dimensional torus of size $18\times 18$. The figure on the right depicts the spherical derivative of the occupation ratio of this torus.
• Figure 4: A contour $\gamma$ in a $10$ by $10$ torus. Vertices $v$ such that $\sigma(v)=1$ are in dark gray and vertices $v$ such that $\sigma(v)=0$ are in white. The shaded gray region denotes the support of $\gamma$. The label of $\mathbb{Z}_{10}^2 \setminus \overline{\gamma}$ is even.
• Figure 5: A matching set of contours in an $18$ by $18$ torus. The contour $\gamma_1$ is small of type even, $\gamma_2$ is small of type odd and $\gamma_3$ is large. The contours $\gamma_2$ and $\gamma_1$ lie in the odd-interior of $\gamma_3$, the contour $\gamma_1$ lies in the even-interior of $\gamma_2$.

Theorems & Definitions (119)

• Proposition 1.1
• Remark 3.1
• Theorem 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Lemma 3.6
• proof
• Remark 3.7
• Lemma 3.8