The deletion channel is the simplest point-to-point communication channel that models lack of synchronization. Despite significant effort, little is known about its capacity, and even less about optimal coding schemes. In this paper we initiate a new systematic approach to this problem, by demonstrating that capacity can be computed in a series expansion for small deletion probability. We compute two leading terms of this expansion, and show that capacity is achieved, up to this order, by i.i.d. uniform random distribution of the input.We think that this strategy can be useful in a number of capacity calculations.

A voter sits on each vertex of an infinite tree of degree $k$, and has to decide between two alternative opinions. At each time step, each voter switches to the opinion of the majority of her neighbors. We analyze this majority process when opinions are initialized to independent and identically distributed random variables. In particular, we bound the threshold value of the initial bias such that the process converges to consensus. In order to prove an upper bound, we characterize the process of a single node in the large $k$-limit. This approach is inspired by the theory of mean field spin-glass and can potentially be generalized to a wider class of models. We also derive a lower bound that is nontrivial for small, odd values of $k$.Comment: Published in at http://dx.doi.org/10.1214/10-AAP729 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The XOR-satisfiability (XORSAT) problem requires finding an assignment of n Boolean variables that satisfy m exclusive OR (XOR) clauses, whereby each clause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing n variables and m clauses of size k. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as k-satisfiability (k-SAT). For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems.We prove a complete characterization of this clustering phase transition for random k-XORSAT. In particular we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold.Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is achieved through a low complexity iterative algorithm.

The deletion channel is the simplest point-to-point communication channel that models lack of synchronization. Input bits are deleted independently with probability d, and when they are not deleted, they are not affected by the channel. Despite significant effort, little is known about the capacity of this channel, and even less about optimal coding schemes. In this paper we develop a new systematic approach to this problem, by demonstrating that capacity can be computed in a series expansion for small deletion probability. We compute three leading terms of this expansion, and find an input distribution that achieves capacity up to this order. This constitutes the first optimal coding result for the deletion channel.The key idea employed is the following: We understand perfectly the deletion channel with deletion probability d = 0. It has capacity 1 and the optimal input distribution is i.i.d. Bernoulli(1/2). It is natural to expect that the channel with small deletion probabilities has a capacity that varies smoothly with d, and that the optimal input distribution is obtained by smoothly perturbing the i.i.d. Bernoulli(1/2) process. Our results show that this is indeed the case. We think that this general strategy can be useful in a number of capacity calculations.whereFurther, the binary stationary source defined by the property that the times at which it switches from 0 to 1 or viceversa form a renewal process with holding time distribution p L (l) = 2 −l (1 + d(l ln l − c 2 l/2)), achieves rate within O(d 3−ǫ ) of capacity.Given a binary sequence, we will call 'runs' its maximal blocks of contiguous 0's or 1's. We shall refer to binary sources such that the switch times form a renewal process as sources (or processes) with i.i.d. runs.The 'rate' of a given binary source is the maximum rate at which information can be transmitted through the deletion channel using input sequences distributed as the source. A formal definition is provided below (see Definition 2.3). Logarithms denoted by log here (and in the rest of the paper) are understood to be in base 2. While one might be skeptical about the concrete meaning of asymptotic expansions of the type (1), they often prove surprisingly accurate. For instance at d = 0.1 (10% of the input symbols are deleted), the expression in Eq. (1) (dropping the error term O(d 3−ǫ )) is larger than the best lower bound [2] by about 0.007 bits. The lower bound of [2] is derived using a Markov source and 'jigsaw' decoding. Our asymptotic analysis implies that the loss in rate due to restricting to Markov sources and jigsaw decoding (cf. Theorem 6.1 and Remark 6.2), to leading order, is 0.904d 2 ≈ 0.009. Hence, we estimate that our asymptotic approach incurs an error of about 0.002 bits for computing the capacity at d = 0.1.More importantly asymptotic expansions can provide useful design insight. Theorem 1.1 shows that the stationary process consisting of i.i.d. runs with the specified run length distribution, achieves capacity to within O(d 3−ǫ ). In comparison, the best performi...

One of the most important challenges in the integration of renewable energy sources into the power grid lies in their 'intermittent' nature. The power output of sources like wind and solar varies with time and location due to factors that cannot be controlled by the provider. Two strategies have been proposed to hedge against this variability: 1) use energy storage systems to effectively average the produced power over time; 2) exploit distributed generation to effectively average production over location. We introduce a network model to study the optimal use of storage and transmission resources in the presence of random energy sources. We propose a Linear-Quadratic based methodology to design control strategies, and we show that these strategies are asymptotically optimal for some simple network topologies. For these topologies, the dependence of optimal performance on storage and transmission capacity is explicitly quantified.