We study the Glauber dynamics for Ising model on (sequences of) dense graphs. We view the dense graphs through the lens of graphons [19]. For the ferromagnetic Ising model with inverse temperature β on a convergent sequence of graphs {G n } with limit graphon W we show fast mixing of the Glauber dynamics if βλ 1 (W) < 1 and slow (torpid) mixing if βλ 1 (W) > 1 (where λ 1 (W) is the largest eigenvalue of the graphon). We also show that in the case βλ 1 (W) = 1 there is insufficient information

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... determine the mixing time (it can be either fast or slow). 1 Introduction Spin systems have been extensively studied in physics [11], mathematics [25], and machine learning [24]. An important and challenging computational question is efficiently sampling configurations from the distribution of a model (spin system). A popular sampling method (and the focus of our paper) is Glauber dynamics [11]. One of the most studied spin models is Ising model [14, 12]. Even though there is a polynomial-time algorithm to sample from the distribution of the ferromagnetic Ising model [13] it is still useful (for reasons of simplicity, generality, and speed) to study the Glauber dynamics for the model [15, 21]. A basic question is: what properties of the underlying graph and the temperature make the Glauber dynamics fast (or slow)? In the case of sparse graphs the dynamics was studied for, for example, Z 2 (see, e.g., [20]), general bounded degree graphs [22], and graphs with bounded connective constant [27, 26]. In the case of dense graphs the dynamics was studied for the complete graph [15] (for more general models on the complete graph, see [6, 2]). Our goal is to understand the impact of the structural properties (analaogously to the connective constant) of the dense graphs and the speed of Glauber dynamics. We will view dense graphs through the lens of graphons [19] and use the notions of free energy of a spin system on a graphon [4]. We give a threshold for the inverse temperature below which Glauber dynamics is rapidly mixing *