Exact goodness-of-fit testing for the Ising model

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness-of-fit test for the finite-lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness-of-fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness-of-fit testing for the Ising model which avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.Comment: 20 page

Critical phenomena in complex networks

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview a wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks. We also discuss strong finite size effects in these systems and highlight open problems and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references, extende

Random walks on graphs: ideas, techniques and results

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects.Comment: LateX file, 34 pages, 13 jpeg figures, Topical Revie

Renormalization: an advanced overview

We present several approaches to renormalization in QFT: the multi-scale analysis in perturbative renormalization, the functional methods \`a la Wetterich equation, and the loop-vertex expansion in non-perturbative renormalization. While each of these is quite well-established, they go beyond standard QFT textbook material, and may be little-known to specialists of each other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added, minor improvements; v3: some changes to sect. 5, refs. adde

Quasi-polynomial mixing of the 2D stochastic Ising model with "plus" boundary up to criticality

We considerably improve upon the recent result of Martinelli and Toninelli on the mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $P$ stochastically dominates the extremal plus phase. An important special case is when $P$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{mix}$ is conjectured to be polynomial in $L$. In [MT] it was shown that for a large enough inverse-temperature $\beta$ and any $\epsilon >0$ there exists $c=c(\beta,\epsilon)$ such that $\lim_{L\to\infty}P(T_{mix}\geq \exp({c L^\epsilon}))=0$. In particular, for the all-plus boundary conditions and $\beta$ large enough $T_{mix} \leq \exp({c L^\epsilon})$. Here we show that the same conclusions hold for all $\beta$ larger than the critical value $\beta_c$ and with $\exp({c L^\epsilon})$ replaced by $L^{c \log L}$ (i.e. quasi-polynomial mixing). The key point is a modification of the inductive scheme of [MT] together with refined equilibrium estimates that hold up to criticality, obtained via duality and random-line representation tools for the Ising model. In particular, we establish new precise bounds on the law of Peierls contours which quantitatively sharpen the Brownian bridge picture established e.g. in [Greenberg-Ioffe (2005)],[Higuchi (1979)],[Hryniv (1998)].Comment: 45 pages, 14 figure

A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)

We propose a family of local CSS stabilizer codes as possible candidates for self-correcting quantum memories in 3D. The construction is inspired by the classical Ising model on a Sierpinski carpet fractal, which acts as a classical self-correcting memory. Our models are naturally defined on fractal subsets of a 4D hypercubic lattice with Hausdorff dimension less than 3. Though this does not imply that these models can be realised with local interactions in 3D Euclidean space, we also discuss this possibility. The X and Z sectors of the code are dual to one another, and we show that there exists a finite temperature phase transition associated with each of these sectors, providing evidence that the system may robustly store quantum information at finite temperature.Comment: 16 pages, 6 figures. In v2, erroneous argument about embeddability into R3 was removed. In v3, minor changes to match journal versio

Cooperative behaviour in complex systems

In my PhD thesis I studied cooperative phenomena arise in complex systems using the methods of statistical and computational physics. The aim of my work was also to study the critical behaviour of interacting many-body systems during their phase transitions and describe their universal features analytically and by means of numerical calculations. In order to do so I completed studies in four different subjects. My first investigated subject was a study of non-equilibrium phase transitions in weighted scale-free networks. The second problem I examined was the ferromagnetic random bond Potts model with large values of q on evolving scale-free networks which problem is equivalent to an optimal cooperation problem. The third examined problem was related to the large-q sate random bond Potts model also and I examined the critical density of clusters which touched a certain border of a perpendicular strip like geometry and expected to hold analytical forms deduced from conformal invariance. The last investigated problem was a study of the non-equilibrium dynamical behaviour of the antiferromagnetic Ising model on two-dimensional triangular lattice at zero temperature in the absence of external field and at the Kosterlitz-Thouless phase transition point.Comment: PhD thesis, 155 pages, 42 figure