Charting the replica symmetric phase

Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous ‘cavity method’, physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007). In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the k-XORSAT model and the diluted k-spin model for even k. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention (Decelle et al. in Phys Rev E 84:066106, 2011)

Strong replica symmetry in high-dimensional optimal Bayesian inference

We consider generic optimal Bayesian inference, namely, models of signal reconstruction where the posterior distribution and all hyperparameters are known. Under a standard assumption on the concentration of the free energy, we show how replica symmetry in the strong sense of concentration of all multioverlaps can be established as a consequence of the Franz-de Sanctis identities; the identities themselves in the current setting are obtained via a novel perturbation coming from exponentially distributed "side-observations" of the signal. Concentration of multioverlaps means that asymptotically the posterior distribution has a particularly simple structure encoded by a random probability measure (or, in the case of binary signal, a non-random probability measure). We believe that such strong control of the model should be key in the study of inference problems with underlying sparse graphical structure (error correcting codes, block models, etc) and, in particular, in the rigorous derivation of replica symmetric formulas for the free energy and mutual information in this context

Broadcasting with Random Matrices

Motivated by the theory of spin-glasses in physics, we study the so-called reconstruction problem for the related distributions on the tree, and on the sparse random graph $G(n,d/n)$. Both cases, reduce naturally to studying broadcasting models on the tree, where each edge has its own broadcasting matrix, and this matrix is drawn independently from a predefined distribution. In this context, we study the effect of the configuration at the root to that of the vertices at distance $h$, as $h\to\infty$. We establish the reconstruction threshold for the cases where the broadcasting matrices give rise to symmetric, 2-spin Gibbs distributions. This threshold seems to be a natural extension of the well-known Kesten-Stigum bound which arises in the classic version of the reconstruction problem. Our results imply, as a special case, the reconstruction threshold for the well-known Edward-Anderson model of spin-glasses on the tree. Also, we extend our analysis to the setting of the Galton-Watson tree, and the random graph $G(n,d/n)$, where we establish the corresponding thresholds.Interestingly, for the Edward-Anderson model on the random graph, we show that the replica symmetry breaking phase transition, established in [Guerra and Toninelli:2004], coincides with the reconstruction threshold. Compared to the classical Gibbs distributions, the spin-glasses have a lot of unique features. In that respect, their study calls for new ideas, e.g., we introduce novel estimators for the reconstruction problem. Furthermore, note that the main technical challenge in the analysis is the presence of (too) many levels of randomness. We manage to circumvent this problem by utilising recently proposed tools coming from the analysis of Markov chains

The ising antiferromagnet and max cut on random regular graphs

The Ising antiferromagnet is an important statistical physics model with close connections to the MAX CUT problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the MAX CUT of random regular graphs predicted by Zdeborová and Boettcher [Journal of Statistical Mechanics 2010]. As an application we prove that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten-Stigum bound

Distributed Storage with Strong Data Integrity based on Blockchain Mechanisms

Master's thesis in Computer scienceA blockchain is a datastructure that is an append-only chain of blocks. Each block contains a set of transaction and has a cryptographic link back to its predecessor. The cryptographic link serves to protect the integrity of the blockchain. A key property of blockchain systems is that it allows mu- tually distrusting entities to reach consensus over a unique order in which transactions are appended. The most common usage of blockchains is in cryptocurrencies such as Bitcoin. In this thesis we use blockchain technology to design a scalable architec- ture for a storage system that can provide strong data integrity and ensure the permanent availability of the data. We study recent literature in blockchain and cryptography to identify the desired characteristics of such a system. In comparison to similar systems, we are able to gain increased performance by designing ours around a permissioned blockchain, allowing only a predefined set of nodes to write to the ledger. A prototype of the system is built on top of existing open-source software. An experimental evaluation using different quorum sizes of the prototype is also presented

Algorithms for #BIS-hard problems on expander graphs

We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) Δ-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite Δ-regular tree