The goal of this paper is to accurately describe the maximal zero-free region of the independence polynomial for graphs of bounded degree, for large degree bounds. In previous work with de Boer, Guerini and Regts it was demonstrated that this zero-free region coincides with the normality region of the related occupation ratios. These ratios form a discrete semi-group that is in a certain sense generated by finitely many rational maps. We will show that as the degree bound converges to infinity, the properly rescaled normality regions converge to a limit domain, which can be described as the maximal boundedness component of a semi-group generated by infinitely many exponential maps.We prove that away from the real axis, this boundedness component avoids a neighborhood of the boundary of the limit cardioid, answering a recent question by Andreas Galanis. We also give an exact formula for the boundary of the boundedness component near the positive real boundary point.
We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.
In this paper we prove that for any integer q ≥ 5, the anti-ferromagnetic q-state Potts model on the infinite ∆-regular tree has a unique Gibbs measure for all edge interaction parameters w ∈ [1 − q/∆, 1), provided ∆ is large enough. This confirms a longstanding folklore conjecture.
We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter
$\lambda $
on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics but have also been key in the recent deterministic approximation scheme for all
$|\lambda |\neq 1$
by Liu, Sinclair and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle and on the tantalising question of what happens around
$\lambda =1$
, where, on one hand, the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability and, on the other hand, the presence of Lee–Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point
$\lambda =1$
and, more generally, on the entire unit circle. For an integer
$\Delta \geq 3$
and edge interaction parameter
$b\in (0,1)$
, we show
$\mathsf {\#P}$
-hardness for approximating the partition function on graphs of maximum degree
$\Delta $
on the arc of the unit circle where the Lee–Yang zeros are dense. This result contrasts with known approximation algorithms when
$|\lambda |\neq 1$
or when
$\lambda $
is in the complementary arc around
$1$
of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee–Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.
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