Let A be a random m × n matrix over the finite field F q with precisely k non-zero entries per row and let y ∈ F m q be a random vector chosen independently of A. We identify the threshold m/n up to which the linear system Ax = y has a solution with high probability and analyse the geometry of the set of solutions. In the special case q = 2, known as the random k-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002 for k = 3, Pittel and Sorkin 2016 for k > 3], and the proof technique was subsequently extended to the cases q = 3,4 [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to q > 3. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.
Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark instances for algorithms and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution [Krzakala et al., PNAS 2007] physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications on Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g., [Banks et al., COLT 2016]).
A cyclic urn is an urn model for balls of types 0, . . . , m − 1. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is j, it is then returned to the urn together with a new ball of type j + 1 mod m. The case m = 2 is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after n steps is, after normalization, known to be asymptotically normal for 2 ≤ m ≤ 6. For m ≥ 7 the normalized composition vector is known not to converge. However, there is an almost sure approximation by a periodic random vector.In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all 7 ≤ m ≤ 12. For m ≥ 13 we also find asymptotically normal fluctuations when normalizing in a more refined way. These fluctuations are of maximal dimension m − 1 only when 6 does not divide m. For m being a multiple of 6 the fluctuations are supported by a two-dimensional subspace.MSC2010: 60F05, 60F15, 60C05, 60J10.
A cyclic urn is an urn model for balls of types 0, . . . , m − 1 where in each draw the ball drawn, say of type j, is returned to the urn together with a new ball of type j + 1 mod m. The case m = 2 is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after n steps is, after normalization, known to be asymptotically normal for 2 ≤ m ≤ 6. For m ≥ 7 the normalized composition vector does not converge. However, there is an almost sure approximation by a periodic random vector. In this paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all m ≥ 7. However, they are of maximal dimension m − 1 only when 6 does not divide m. For m being a multiple of 6 the fluctuations are supported by a two-dimensional subspace.MSC2010: 60F05, 60F15, 60C05, 60J10.
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