Renormalization group analysis of the Anderson model on random regular graphs

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

Une permutation donnée, vue comme réunion de cycles disjoints, représente un graphe régulier de degré $2$. Considérons $d$ permutations aléatoires indépendantes, en superposons leurs structures de graphes. Ceci est le modèle de permutation bien connu donnant un (multi-)graphe régulier aléatoire de degré $2d$. Nous considérons un champ $2$-dimensionnel de $d$ permutations indexé par la taille et le temps. La taille de chaque permutation croît selon des processus couplés de restaurants chinois, tandis que chaque permutation évolue dans le temps selon une chaîne de transpositions aléatoires. À travers le modèle de permutation, ceci se projette en une famille à deux paramètres de graphes qui croissent en taille (« dimension ») et qui évoluent en temps. Dans cet ensemble de graphes aléatoires, on observe asymptotiquement une évolution remarquable des petits cycles et des statistiques linéaires des valeurs propres en dimension et en temps. En dimension, il avait été montré par Johnson et Pal (Ann. Probab. 42 (2014) 1396–1437) que les nombres de cycles sont décrits par un champ poissonnien de processus de Yule. Ici, nous donnons une description en dimension et en temps du processus des cycles en termes en termes d’un surface aléatoire poissonnienne, pour tout $d$. Lorsque $d$ tend vers l’infini, les fluctuations des nombres de cycles convergent vers un processus gaussien indexé par la dimension et le temps. Les marginales en dimension se trouvent être le champ libre gaussien, et le processus est stationnaire en temps. Une structure similaire de covariance apparaît dans les fluctuations des valeurs propres du processus des mineurs d’une matrice de Wigner réelle symétrique dont les coordonnées évoluent selon des processus stochastiques stationnaires i.i.d.. Ainsi, cet article décrit un analogue poissonnien d’une dynamique markovienne naturelle sur le champ libre gaussien, et étudie ses propriétés trajectorielles.

Let G be a given graph (modelling a communication network) which we assume suffers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 - p). In particular we are interested in robustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our result is: If the degree d is fixed then p = 1/(d - 1) is a threshold probability for the existence of a linear-sized component in a faulty version of almost all random regular graphs. We show: If each edge of an arbitrary graph G with maximum degree bounded above by d is present with, probability p = Λ/(d - 1) where Λ 1 then for almost all G the faulted version of G has a linear-sized component with high probability. Note that these results imply some kind of optimality of random regular graphs among the class of graphs with the same degree bound. The theme is: Use the known expansion properties of almost all random regular graphs to obtain strong robustness results. This has not been done systematically before.

We perform a renormalization group (RG) analysis of collinear hadron production in deep inelastic scattering on nuclei. We consider the limit where the parent parton energy E is large, while the medium opacity L/λg remains small. We identify the fixed order and leading ln⁡(E/ξ2L) enhanced medium contributions to the semi-inclusive cross sections and derive RG equations that resum multiple emissions near the endpoints of the splitting functions at first order in opacity. These evolution equations treat the same type of radiation enhancement in matter as the modified Dokshitzer-Gribov-Lipatov-Altarelli-Parisi approach, but differ in the way one regulates the collinear divergences. They provide a unique analytic insight into the problem of resummation and a faster and more efficient path to phenomenology. The new RG evolution framework is applied to study fragmentation in eA reactions.

We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.

We introduce a general class of algorithms and analyse their application to regular graphs of large girth. In particular, we can transfer several results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This reverses the usual direction, which is from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As a result, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum k-independent sets, minimum k-dominating sets and maximum k-separated matchings in r-regular graphs with large girth.

We study the return probability for the Anderson model on the random regular graph and give evidence of the existence of two distinct phases: a fully ergodic and nonergodic one. In the ergodic phase, the return probability decays polynomially with time with oscillations, being the attribute of the Wigner-Dyson-like behavior, while in the nonergodic phase the decay follows a stretched exponential decay.We give a phenomenological interpretation of the stretched exponential decay in terms of a classical random walker. Furthermore, comparing typical and mean values of the return probability, we show how to differentiate an ergodic phase from a nonergodic one. We benchmark this method first in two random matrix models, the power-law random banded matrices, and the Rosenzweig-Porter matrices, which host both phases. Second, we apply this method to the Anderson model on the random regular graph to give further evidence of the existence of the two phases.

The role of cooperative effects (i.e. synergy) in transmission of infection is investigated analytically and numerically for epidemics following the rules of susceptible-infected-susceptible (SIS) model defined on random regular graphs. Non-linear dynamics are shown to lead to bifurcation diagrams for such spreading phenomena exhibiting three distinct regimes: non-active, active and bi-stable. The dependence of bifurcation loci on node degree is studied and interesting effects are found that contrast with the behaviour expected for non-synergistic epidemics.

Several popular problems in information networking and social networking such as search, distribution, and dissemination can be modeled by random walk of single or multiple agents on a graph. As an approach to remedy the redundancy in multiple random walks, Branching Random Walk (BRW) has been proposed, in which the number of agents of random walks is dynamically changed by the branching-and-aggregation mechanism. An agent starting its walk from an originating node randomly selects b nodes among its neighbor nodes, and it activates those neighbor nodes (branching). If multiple agents simultaneously visit the same node, those agents are unified into a single agent (aggregation). BRW is a relatively new mobility model on a graph. Hence, to the best of our knowledge, understanding of its dynamical properties is quite limited. In this paper, we describe the dynamics of BRW on a random regular graph, and clarify the relation between the branching parameter b and the mean and the distribution of hitting times and the mean cover time.

According to Landau criterion, a phase transition should be first order when cubic terms of order parameters are allowed in its effective Ginzburg-Landau free energy. Recently, it was shown by renormalization group (RG) analysis that continuous transition can happen at putatively first-order $Z_3$ transitions in 2D Dirac semimetals and such non-Landau phase transitions were dubbed "fermion-induced quantum critical points" (FIQCP) [Li et al., Nature Communications 8, 314 (2017)]. The RG analysis, controlled by the 1/$N$ expansion with $N$ the number of flavors of four-component Dirac fermions, shows that FIQCP occurs for $N\geq N_c$. Previous QMC simulations of a microscopic model of SU($N$) fermions on the honeycomb lattice showed that FIQCP occurs at the transition between Dirac semimetals and Kekule-VBS for $N\geq 2$. However, precise value of the lower bound $N_c$ has not been established. Especially, the case of $N=1$ has not been explored by studying microscopic models so far. Here, by introducing a generalized SU($N$) fermion model with $N=1$ (namely spinless fermions on the honeycomb lattice), we perform large-scale sign-problem-free Majorana quantum Monte Carlo simulations and find convincing evidence of FIQCP for $N=1$. Consequently, our results suggest that FIQCP can occur in 2D Dirac semimetals for all positive integers $N\geq 1$.

Motivated by the problem of many-body localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig–Porter random matrix model is suggested that possesses two transitions. One of them is the Anderson localization transition from the localized to the extended states. The other one is the ergodic transition from the extended non-ergodic (multifractal) states to the extended ergodic states. We confirm the existence of both transitions by computing the two-level spectral correlation function, the spectrum of multifractality and the wave function overlap which consistently demonstrate these two transitions.

Quasirandom Rumor Spreading on Expanders

Network sampling is an indispensable tool for understanding features of large complex networks where it is practically impossible to search over the entire graph. In this paper, we develop a framework for statistical inference for counting network motifs, such as edges, triangles and wedges, in the widely used subgraph sampling model, where each vertex is sampled independently, and the subgraph induced by the sampled vertices is observed. We derive necessary and sufficient conditions for the consistency and the asymptotic normality of the natural Horvitz–Thompson (HT) estimator, which can be used for constructing confidence intervals and hypothesis testing for the motif counts based on the sampled graph. In particular, we show that the asymptotic normality of the HT estimator exhibits an interesting fourth-moment phenomenon, which asserts that the HT estimator (appropriately centered and rescaled) converges in distribution to the standard normal whenever its fourth-moment converges to 3 (the fourth-moment of the standard normal distribution). As a consequence, we derive the exact thresholds for consistency and asymptotic normality of the HT estimator in various natural graph ensembles, such as sparse graphs with bounded degree, Erdős–Rényi random graphs, random regular graphs and dense graphons.

OF THE DISSERTATION Random matrices and random boxes by Linh V. Tran Dissertation Director: Prof. Van Vu This thesis concerns two questions on random structures: the semi-circular law for adjacency matrix of regular random graph and the piercing number for random boxes. • Random matrices We proved in full generality the semi-circular law for random d-regular graph model in the case d tends to infinity as n does. Our result complements the McKay law [19], which applied for the case d is an absolute constant. • Random boxes. Take n random boxes with axis-parallel edges inside the unit cube [0, 1]d, the piercing number τd(n) is the minimum number of points needed to pierce all boxes. Using hypergraph setting, we was able to prove a near sharp estimation for τd(n): for any d ≥ 2 Ωd( √ n(log n)d/2−1) = τd(n) = Od( √ n(log n)d/2−1 log log n). This thesis is based on two papers by the author [31] and [30] (joint work with Van Vu and Ke Wang).

Since today’s HPC and data center systems can comprise hundreds of thousands of servers and beyond, it is crucial to equip them with a network that provides high performance. New topologies proposed to achieve such performance need to be evaluated under different traffic conditions, aiming to closely replicate real-world scenarios. While most optimizations should be guided by common traffic patterns, it is essential to ensure that no pathological traffic pattern can compromise the entire system. Determining synthetic adversarial traffic patterns for a network typically relies on a thorough understanding of its topology and routing. In this paper, we address the problem of identifying a generic adversarial traffic pattern for low-diameter direct interconnection networks. We first focus on Random Regular Graphs (RRGs), which represent a typical case for these networks. Moreover, RRGs have been proposed as topologies for interconnection networks due to their superior scalability and expandability, among other advantages. We introduce Ant Mill, an adversarial traffic pattern for RRGs when using routes of minimal length. Secondly, we demonstrate that the Ant Mill traffic pattern is also adversarial in other low-diameter direct interconnection networks such as Slimfly, Dragonfly, and Projective networks. Ant Mill is thoroughly motivated and evaluated, enabling future studies of low-diameter direct interconnection networks to leverage its findings.