We give a complete characterization of the two-state anti-ferromagnetic spin systems which are of strong spatial mixing on general graphs. We show that a two-state anti-ferromagnetic spin system is of strong spatial mixing on all graphs of maximum degree at most ∆ if and only if the system has a unique Gibbs measure on infinite regular trees of degree up to ∆, where ∆ can be either bounded or unbounded. As a consequence, there exists an FPTAS for the partition function of a two-state anti-ferromagnetic spin system on graphs of maximum degree at most ∆ when the uniqueness condition is satisfied on infinite regular trees of degree up to ∆. In particular, an FPTAS exists for arbitrary graphs if the uniqueness is satisfied on all infinite regular trees. This covers as special cases all previous algorithmic results for two-state anti-ferromagnetic systems on general-structure graphs.Combining with the FPRAS for two-state ferromagnetic spin systems of Jerrum-Sinclair and Goldberg-Jerrum-Paterson, and the very recent hardness results of Sly-Sun and independently of Galanis-Stefankovic-Vigoda, this gives a complete classification, except at the phase transition boundary, of the approximability of all two-state spin systems, on either degree-bounded families of graphs or family of all graphs.

Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength W , where a quantum to classical crossover at time tc ∝ W −2 transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model (which is defined as a weighted sum over all matchings where each matching is given a weight γ |V |−2|M | in terms of a fixed parameter γ called the monomer activity) and the hard core model (which is defined as a weighted sum over all independent sets where an independent set I is given a weight λ |I| in terms of a fixed parameter λ called the vertex activity). The connective constant is a natural measure of the average degree of a graph which has been studied extensively in combinatorics and mathematical physics, and can be bounded by a constant even for certain unbounded degree graphs such as those sampled from the sparse Erdős-Rényi model G(n, d/n).Our main technical contribution is to prove the best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant. These results on decay of correlations are obtained using a new framework based on the so-called message approach that has been extensively used recently to prove such results for bounded degree graphs. We then use these optimal decay of correlations results to obtain FPTASs for the two problems on graphs of bounded connective constant.In particular, for the monomer-dimer model, we give a deterministic FPTAS for the partition function on all graphs of bounded connective constant for any given value of the monomer activity. The best previously known deterministic algorithm was due to Bayati, Gamarnik, Katz, Nair and Tetali [STOC 2007], and gave the same runtime guarantees as our results but only for the case of bounded degree graphs. For the hard core model, we give an FPTAS for graphs of connective constant Δ whenever

We give the first deterministic fully polynomial-time approximation scheme (FPTAS) for computing the partition function of a two-state spin system on an arbitrary graph, when the parameters of the system satisfy the uniqueness condition on infinite regular trees. This condition is of physical significance and is believed to be the right boundary between approximable and inapproximable.The FPTAS is based on the correlation decay technique introduced by Bandyopadhyay and Gamarnik [SODA 06] and Weitz [STOC 06]. The classic correlation decay is defined with respect to graph distance. Although this definition has natural physical meanings, it does not directly support an FPTAS for systems on arbitrary graphs, because for graphs with unbounded degrees, the local computation that provides a desirable precision by correlation decay may take super-polynomial time. We introduce a notion of computationally efficient correlation decay, in which the correlation decay is measured in a refined metric instead of graph distance. We use a potential method to analyze the amortized behavior of this correlation decay and establish a correlation decay that guarantees an inverse-polynomial precision by polynomial-time local computation. This gives us an FPTAS for spin systems on arbitrary graphs. This new notion of correlation decay properly reflects the algorithmic aspect of the spin systems, and may be used for designing FPTAS for other counting problems.

The hard core model in statistical physics is a probability distribution on independent sets in a graph in which the weight of any independent set I is proportional to λ |I| , where λ > 0 is the vertex activity. We show that there is an intimate connection between the connective constant of a graph and the phenomenon of strong spatial mixing (decay of correlations) for the hard core model; speci cally, we prove that the hard core model with vertex activity λ < λ c (∆ + 1) exhibits strong spatial mixing on any graph of connective constant ∆, irrespective of its maximum degree, and hence derive an FPTAS for the partition function of the hard core model on such graphs. Here λ c (d) · · = d d (d−1) d+1 is the critical activity for the uniqueness of the Gibbs measure of the hard core model on the in nite d-ary tree. As an application, we show that the partition function can be e ciently approximated with high probability on graphs drawn from the random graph model G (n, d/n) for all λ < e/d, even though the maximum degree of such graphs is unbounded with high probability.We also improve upon Weitz's bounds for strong spatial mixing on bounded degree graphs [32] by providing a computationally simple method which uses known estimates of the connective constant of a lattice to obtain bounds on the vertex activities λ for which the hard core model on the lattice exhibits strong spatial mixing. Using this framework, we improve upon these bounds for several lattices including the Cartesian lattice in dimensions 3 and higher.Our techniques also allow us to relate the threshold for the uniqueness of the Gibbs measure on a general tree to its branching factor [17

An asynchronous algorithm is described for rapidly constructing an overlay network in a peer-to-peer system where all nodes can in principle communicate with each other directly through an underlying network, but each participating node initially has pointers to only a handful of other participants. The output of the mechanism is a linked list of all participants sorted by their identifiers, which can be used as a foundation for building various linear overlay networks such as Chord or skip graphs. Assuming the initial pointer graph is weakly-connected with maximum degree d and the length of a node identifier is W , the mechanism constructs a binary search tree of nodes of depth O(W ) in expected O(W log n) time using expected O((d + W )n log n) messages of size O(W ) each. Furthermore, the algorithm has low contention: at any time there are only O(d) undelivered messages for any given recipient. A lower bound of Ω(d + log n) is given for the running time of any procedure in a related synchronous model that yields a sorted list from a degree-d weakly-connected graph of n nodes. We conjecture that this lower bound is tight and could be attained by further improvements to our algorithms.

The local computation of Linial [FOCS'87] and Naor and Stockmeyer [STOC'93] concerns with the question of whether a locally definable distributed computing problem can be solved locally: more specifically, for a given local CSP whether a CSP solution can be constructed by a distributed algorithm using local information. In this paper, we consider the problem of sampling a uniform CSP solution by distributed algorithms, and ask whether a locally definable joint distribution can be sampled from locally. More broadly, we consider sampling from Gibbs distributions induced by weighted local CSPs, especially the Markov random fields (MRFs), in the LOCAL model.We give two Markov chain based distributed algorithms which we believe to represent two fundamental approaches for sampling from Gibbs distributions via distributed algorithms. The first algorithm generically parallelizes the single-site sequential Markov chain by iteratively updating a random independent set of variables in parallel, and achieves an O(∆ log n) time upper bound in the LOCAL model, where ∆ is the maximum degree, when the Dobrushin's condition for the Gibbs distribution is satisfied. The second algorithm is a novel parallel Markov chain which proposes to update all variables simultaneously yet still guarantees to converge correctly with no bias. It surprisingly parallelizes an intrinsically sequential process: stabilizing to a joint distribution with massive local dependencies, and may achieve an optimal O(log n) time upper bound independent of the maximum degree ∆ under a stronger mixing condition.We also show a strong Ω(diam) lower bound for sampling: in particular for sampling independent set in graphs with maximum degree ∆ ≥ 6. Independent sets are trivial to construct locally and the sampling lower bound holds even when every node is aware of the entire graph. This gives a strong separation between sampling and constructing locally checkable labelings.

Glauber theory can describe elastic scattering of a. particles by 4 He, 3 He, 2 H, and *H at 7 GeV/c if the phase of the nucleon-nucleon elastic-scattering amplitude varies with momentum transfer. The phase variation leads to diffraction patterns differing markedly from those typical of constant-phase calculations and greatly affects the magnitudes of the intensities. These changes are mainly due to changes in the interference between amplitudes for different orders of multiple scattering and to a decrease in their moduli. PACS numbers: 25.55.Ci, 21.30. + y, 24.10.Ht During the past twenty years the Glauber theory has been extremely successful in describing hadronnucleus elastic scattering at energies of approximately 1 GeV or higher. This success has not been shared to the same degree in nucleus-nucleus ("heavy-ion") elastic scattering at corresponding energies of 1 GeV/nucleon or higher for several reasons. First, there has been a relative paucity of such measurements. Second, the extension of the theory to nucleus-nucleus collisions is significantly more complex and the computations are more difficult and lengthy so that fewer of these types of calculations exist. 1 Recently a comprehensive set of measurements of elastic scattering of a particles by four very light nuclei ( 4 He, 3 He, 2 H, *H) was made at an incident a-particle momentum of 7 GeV/c over a range of |r| values from -0.07 to -4 (GeV/c) 2 . The cross sections fell from the barn to the nanobarn level. Such data, in which the intensities vary through so many orders of magnitude and over such a large range of momentum transfers, are extremely useful because they put enormous constraints on any theory. It is no longer sufficient to show that the theory describes measurements of collisions between just one given pair of nuclei. Now the theory must describe measurements between four different pairs of nuclei, and it must do so consistently. Whatever nucleon-nucleon (AW) elasticscattering amplitude is used for one calculation should be used for the others as well. In addition, since these measurements have gone out to rather large momentum transfers, the calculated intensities will be much more sensitive to the AW elastic scattering amplitudes used as input.The measurements for elastic scattering of a particles by the four light nuclei were accompanied 2 by theoretical analyses for the a-2 H, «-3 He, and «-4 He cross sections. These analyses were both by means of the so-called "rigid projectile approximation" and by means of the Glauber theory, with Gaussian densities for the nuclear ground states. The rigid-projectile approximation failed even qualitatively except at very small momentum transfers. 2 In the Glauber-theory calculations shown, 2 the broad qualitative trends of the data were to some extent roughly described. Quantitatively the results were in strong disagreement with the data, often being as much as an order of magnitude too low.In the present analysis we have calculated the elastic-scattering differential cross sections for all four p...