We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation @math is proportional to @math where @math is a positive parameter and @math is the number of inversions in @math . In our main result we show that when @math , the limiting distribution of the longest increasing subsequence (LIS) is Gaussian, answering an open question in [Bhatnagar and Peled, PTRF, 2015]. This is in contrast to the case when @math where the limiting distribution of the LIS when scaled appropriately is the GUE Tracy-Widom distribution. We also obtain a law of large numbers for the length of the longest decreasing subsequence (LDS) and identify the precise constant in the order of the expectation, answering a further open question in [Bhatnagar and Peled, PTRF, 2015].
When @math , the integrable structure is lost, and the powerful combinatorial, algebraic and analytic tools used in @cite_2 are no longer available for finding the limiting distribution of @math . Indeed, as Theorem shows, for @math bounded away from @math , we get a different scaling limit, namely Gaussian with diffusive scaling, in contrast with the Tracy-Widom limit with subdiffusive scaling one gets for @math .
@cite_2
2114789426 1992413315 2115976335 2127090196
We consider the problem of robustly recovering a @math -sparse coefficient vector from the Fourier series that it generates, restricted to the interval @math . The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of @math ) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm @math , we show upper and lower bounds on the minimax error rate that both scale like @math , providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity @math , or equivalently from comparing @math to the so-called @math -spark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis. We study the scaling limits of three different aggregation models on ℤd: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in ℝd. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains. We flnd scaling limits for the sizes of the largest components at criticality for the rank-1 inho- mogeneous random graphs with power-law degrees with exponent ?. We investigate the case where ? 2 (3;4), so that the degrees have flnite variance but inflnite third moment. The sizes of the largest clusters, rescaled by n i(?i2)=(?i1) , converge to hitting times of a 'thinned' Levy process. This process is intimately connected to the general multiplicative coalescents studied in (1) and (2). In particular, we use the results in (2) to show that, when interpreting the location ‚ inside the critical window as time, the limiting process is a multiplicative process with difiusion constant 0 and the entrance boundary describing the size of relative components in the ‚ ! i1 regime proportional to i i i1=(?i1) ¢ i‚1 . A crucial ingredient is the identiflcation of the scaling of the largest connected components in the barely subcritical regime. Our results should be contrasted to the case where the degree exponent ? satisfles ? > 4, so that the third moment is flnite. There, instead, we see that the sizes of the components rescaled by n i2=3 converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in (1) for the Erd os- Renyi random graph and extended to the present setting in (3, 4). The limit again is a multiplicative coalescent, the only difierence with the limit for ? 2 (3;4) being the initial state, corresponding to the barely subcritical regime. We introduce a new approach to characterizing the unobserved portion of a distribution, which provides sublinear--sample estimators achieving arbitrarily small additive constant error for a class of properties that includes entropy and distribution support size. Additionally, we show new matching lower bounds. Together, this settles the longstanding question of the sample complexities of these estimation problems, up to constant factors. Our algorithm estimates these properties up to an arbitrarily small additive constant, using O(n log n) samples, where n is a bound on the support size, or in the case of estimating the support size, 1 n is a lower bound on the probability of any element of the domain. Previously, no explicit sublinear--sample algorithms for either of these problems were known. Our algorithm is also computationally extremely efficient, running in time linear in the number of samples used. In the second half of the paper, we provide a matching lower bound of Ω(n log n) samples for estimating entropy or distribution support size to within an additive constant. The previous lower-bounds on these sample complexities were n 2O(√log n). To show our lower bound, we prove two new and natural multivariate central limit theorems (CLTs); the first uses Stein's method to relate the sum of independent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance metric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central limit theorem for "generalized multinomial" distributions---a large class of discrete distributions, parameterized by matrices, that represents sums of independent binomial or multinomial distributions, and describes many distributions encountered in computer science. Convergence here is in the strong sense of statistical distance, which immediately implies that any algorithm with input drawn from a generalized multinomial distribution behaves essentially as if the input were drawn from a discretized Gaussian with the same mean and covariance. Such tools in the multivariate setting are rare, and we hope this new tool will be of use to the community.