| Özet: | <p>This thesis investigates fundamental problems in approximate counting that arise in the field of statistical mechanics. Building upon recent advancements in the area, our research aims to enhance our understanding of the computational complexity of sampling from the Ising and Potts models, as well as the random $k$-SAT model.</p>
<p>The $q$-state Potts model is a spin model in which each particle is randomly assigned a spin (out of $q$ possible spins), where the probability of a certain assignment depends on how many adjacent particles present the same spin. The edge interaction of the model is a parameter that quantifies the strength of interaction between two adjacent particles. The Ising model corresponds to the Potts model with $q = 2$. Sampling from these models is inherently connected to approximating the partition function of the model, a graph polynomial that encodes several aggregate thermodynamic properties of the system. In addition to classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane of these partition functions, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Thus, following this trend in both statistical physics and algorithmic research, we allow the edge interaction to be any complex number.</p>
<p>First, we study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Previous work in the complex plane by Goldberg and Guo focused on $q=2$; for $q>2$, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all $q\geq 2$ and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations.</p>
<p>Secondly, we investigate the complexity of approximating the partition function $\ising(G; \beta)$ of the Ising model in terms of the relation between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on the maximum degree of the input graph $G$. In this thesis we establish both new tractability and inapproximability results. Our tractability results show that $\ising(-; \beta)$ has an FPTAS when $\beta \in \mathbb{C}$ and $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert < \tan(\pi / (4 \Delta - 4))$. The core of the proof is showing that there are no inputs~$G$ that make the partition function $0$ when $\beta$ is in this range. Our result significantly extends the known zero-free region of the Ising model (and hence the known approximation results). Our intractability results show that it is $\numP$-hard to approximate $\ising(-; \beta)$ when $\beta \in \mathbb{C}$ is an algebraic number such that $\beta \not \in \mathbb{R} \cup \{i, -i\}$ and $\lvert \beta - 1\rvert / \lvert \beta + 1 \rvert > 1 / \sqrt{\Delta - 1}$. These are the first results to show intractability of approximating $\ising(-, \beta)$ on bounded degree graphs with complex $\beta$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model.</p>
<p>Finally, we exploit the recently successful framework of spectral independence to analyse the mixing time of a Markov chain, and we apply it in order to sample satisfying assignments of $k$-CNF formulas. Our analysis leads to a nearly linear-time algorithm to approximately sample satisfying assignments in the random $k$-SAT model when the density of the random formula $\alpha=m/n$ scales exponentially with $k$, where $n$ is the number of variables and $m$ is the number of clauses. The best previously known sampling algorithm for the random $k$-SAT model applies when the density $\alpha=m/n$ of the formula is less than $2^{k/300}$ and runs in time $n^{\exp(\Theta(k))}$. Our algorithm achieves a significantly faster running time of $n^{1 + o_k(1)}$ and samples satisfying assignments up to density $\alpha\leq 2^{0.039 k}$. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting.</p>
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