Universal Loop-Free Super-Stabilization

We propose an univesal scheme to design loop-free and super-stabilizing protocols for constructing spanning trees optimizing any tree metrics (not only those that are isomorphic to a shortest path tree). Our scheme combines a novel super-stabilizing loop-free BFS with an existing self-stabilizing spanning tree that optimizes a given metric. The composition result preserves the best properties of both worlds: super-stabilization, loop-freedom, and optimization of the original metric without any stabilization time penalty. As case study we apply our composition mechanism to two well known metric-dependent spanning trees: the maximum-flow tree and the minimum degree spanning tree

Disconnected components detection and rooted shortest-path tree maintenance in networks

International audienceMany articles deal with the problem of maintaining a rooted shortest-path tree. However, after some edge deletions, some nodes can be disconnected from the connected component $V_r$ of some distinguished node $r$. In this case, an additional objective is to ensure the detection of the disconnection by the nodes that no longer belong to $V_r$. We present a detailed analysis of a silent self-stabilizing algorithm. We prove that it solves this more demanding task in anonymous weighted networks with the following additional strong properties: it runs without any knowledge on the network and under the \emph{unfair} daemon, that is without any assumption on the asynchronous model. Moreover, it terminates in less than $2n+D$ rounds for a network of $n$ nodes and hop-diameter $D$

Sandpile models

This survey is an extended version of lectures given at the Cornell Probability Summer School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and its connections to related models are presented. We discuss exactly computable results via Majumdar and Dhar's method. The main ideas of Priezzhev's computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice. We also discuss various questions arising on infinite graphs, such as convergence to a sandpile measure, and stabilizability of infinite configurations.Comment: 72 pages - v3 incorporates referee's comments. References closely related to the lectures were added/update

Primer for the algebraic geometry of sandpiles

The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the Laplacian matrix, drawing out connections with the ASM. An extended summary of the ASM and of the required algebraic geometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new construction of arithmetically Gorenstein ideals; a generalization to directed multigraphs of a duality theorem between elements of the sandpile group of a graph and the graph's superstable configurations (parking functions); and a characterization of the top Betti number of the minimal free resolution of the Laplacian lattice ideal as the number of elements of the sandpile group of least degree. A characterization of all the Betti numbers is conjectured.Comment: 45 pages, 14 figures. v2: corrected typo

Dynamical Fractional Chaotic Inflation -- Dynamical Generation of a Fractional Power-Law Potential for Chaotic Inflation

Chaotic inflation based on a simple monomial scalar potential, V(phi) ~ phi^p, is an attractive large-field model of inflation capable of generating a sizable tensor-to-scalar ratio r. Therefore, assuming that future CMB observations will confirm the large r value reported by BICEP2, it is important to determine what kind of dynamical mechanism could possibly endow the inflaton field with such a simple effective potential. In this paper, we answer this question in the context of field theory, i.e. in the framework of dynamical chaotic inflation (DCI), where strongly interacting supersymmetric gauge dynamics around the scale of grand unification dynamically generate a fractional power-law potential via the quantum effect of dimensional transmutation. In constructing explicit models, we significantly extend our previous work, as we now consider a large variety of possible underlying gauge dynamics and relax our conditions on the field content of the model. This allows us to realize almost arbitrary rational values for the power p in the inflaton potential. The present paper may hence be regarded as a first step towards a more complete theory of dynamical chaotic inflation.Comment: 68 pages, 7 figures, 2 tables, 2 appendice

Polynomial Silent Self-Stabilizing p-Star Decomposition

We present a silent self-stabilizing distributed algorithm computing a maximal p-star decomposition of the underlying communication network. Under the unfair distributed scheduler, the most general scheduler model, the algorithm converges in at most 12∆m + O(m + n) moves, where m is the number of edges, n is the number of nodes, and ∆ is the maximum node degree. Regarding the move complexity, our algorithm outperforms the previously known best algorithm by a factor of ∆. While the round complexity for the previous algorithm was unknown, we show a 5 [n/(p+1)] + 5 bound for our algorithm

Pati-Salam Axion

I discuss the implementation of the Peccei-Quinn mechanism in a minimal realization of the Pati-Salam partial unification scheme. The axion mass is shown to be related to the Pati-Salam breaking scale and it is predicted via a two-loop renormalization group analysis to be in the window $m_a \in [10^{-11}, \, 3 \times 10^{-7}]$ eV, as a function of a sliding Left-Right symmetry breaking scale. This parameter space will be fully covered by the late phases of the axion Dark Matter experiments ABRACADABRA and CASPEr-Electric. A Left-Right symmetry breaking scenario as low as 20 TeV is obtained for a Pati-Salam breaking of the order of the reduced Planck mass.Comment: 33 pages, 3 figures. Minor corrections. Matches version to appear in JHE

Chip-Firing and Rotor-Routing on Directed Graphs

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.Comment: 34 pages, 11 figures. v2 has additional references, v3 corrects figure 9, v4 corrects several typo