135,085 research outputs found

    Thresholds and expectation-thresholds of monotone properties with small minterms

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    Let NN be a finite set, let p(0,1)p \in (0,1), and let NpN_p denote a random binomial subset of NN where every element of NN is taken to belong to the subset independently with probability pp . This defines a product measure μp\mu_p on the power set of NN, where for A2N\mathcal{A} \subseteq 2^N μp(A):=Pr[NpA]\mu_p(\mathcal{A}) := Pr[N_p \in \mathcal{A}]. In this paper we study upward-closed families A\mathcal{A} for which all minimal sets in A\mathcal{A} have size at most kk, for some positive integer kk. We prove that for such a family μp(A)/pk\mu_p(\mathcal{A}) / p^k is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables to identify in A\mathcal{A} either a substantial subfamily A0\mathcal{A}_0 for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than kk. Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using duality of linear programming. This is related to the threshold conjecture of Kahn and Kalai

    Rainbow Thresholds

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    We extend a recent breakthrough result relating expectation thresholds and actual thresholds to include rainbow versions

    Threshold for Steiner triple systems

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    We prove that with high probability G(3)(n,n1+o(1))\mathbb{G}^{(3)}(n,n^{-1+o(1)}) contains a spanning Steiner triple system for n1,3(mod6)n\equiv 1,3\pmod{6}, establishing the tight exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.Comment: 22 pages, 1 figur

    Effects on orientation perception of manipulating the spatio–temporal prior probability of stimuli

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    Spatial and temporal regularities commonly exist in natural visual scenes. The knowledge of the probability structure of these regularities is likely to be informative for an efficient visual system. Here we explored how manipulating the spatio–temporal prior probability of stimuli affects human orientation perception. Stimulus sequences comprised four collinear bars (predictors) which appeared successively towards the foveal region, followed by a target bar with the same or different orientation. Subjects' orientation perception of the foveal target was biased towards the orientation of the predictors when presented in a highly ordered and predictable sequence. The discrimination thresholds were significantly elevated in proportion to increasing prior probabilities of the predictors. Breaking this sequence, by randomising presentation order or presentation duration, decreased the thresholds. These psychophysical observations are consistent with a Bayesian model, suggesting that a predictable spatio–temporal stimulus structure and an increased probability of collinear trials are associated with the increasing prior expectation of collinear events. Our results suggest that statistical spatio–temporal stimulus regularities are effectively integrated by human visual cortex over a range of spatial and temporal positions, thereby systematically affecting perception

    Some results on fractional vs. expectation thresholds

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    A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. Expectation (resp. fractional expectation) threshold qq (resp. qfq_f) for an increasing nontrivial class F2X\mathcal{F}\subseteq 2^X allows to locate the threshold for F\mathcal{F} within a logarithmic factor (these are important breakthrough results of Park and Pham (2022), resp. Frankston, Kahn, Narayanan and Park (2019)). We will survey what is known about the relation between qq and qfq_f and prove some further special cases of Talagrand's conjecture.Comment: 17 pages, 0 figure

    Random geometric complexes

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    We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

    Exploiting exotic LHC datasets for long-lived new particle searches

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    Motivated by the expectation that new physics may manifest itself in the form of very heavy new particles, most of the operation time of the LHC is devoted to pppp collisions at the highest achievable energies and collision rates. The large collision rates imply tight trigger requirements that include high thresholds on the final-state particles' transverse momenta pTp_{T} and an intrinsic background in the form of particle pileup produced by different collisions occurring during the same bunch crossing. This strategy is potentially sub-optimal for several well-motivated new physics models where new particles are not particularly heavy and can escape the online selection criteria of the multi-purpose LHC experiments due to their light mass and small coupling. A solution may be offered by complementary datasets that are routinely collected by the LHC experiments. These include heavy ion collisions, low-pileup runs for precision physics, and the so-called 'parking' and 'scouting' datasets. While some of them are motivated by other physics goals, they all have the usage of mild pTp_{T} thresholds at the trigger-level in common. In this study, we assess the relative merits of these datasets for a representative model whose particular clean signature features long-lived resonances yielding displaced dimuon vertices. We compare the reach across those datasets for a simple analysis, simulating LHC data in Run 2 and Run 3 conditions with the Delphes simulation. We show that the scouting and parking datasets, which afford low-pTp_{T} trigger thresholds by only using partial detector information and delaying the event reconstruction, respectively, have a reach comparable to the standard pppp dataset with conventional thresholds. We also show that heavy ion and low-pileup datasets are far less competitive for this signature.Comment: 23 pages with tables and figure
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