135,085 research outputs found
Thresholds and expectation-thresholds of monotone properties with small minterms
Let be a finite set, let , and let denote a random
binomial subset of where every element of is taken to belong to the
subset independently with probability . This defines a product measure
on the power set of , where for
.
In this paper we study upward-closed families for which all
minimal sets in have size at most , for some positive integer
. We prove that for such a family 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
either a substantial subfamily 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 .
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
We extend a recent breakthrough result relating expectation thresholds and
actual thresholds to include rainbow versions
Threshold for Steiner triple systems
We prove that with high probability
contains a spanning Steiner triple system for ,
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
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
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
(resp. ) for an increasing nontrivial class
allows to locate the threshold for 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 and and prove some further special cases of
Talagrand's conjecture.Comment: 17 pages, 0 figure
Random geometric complexes
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
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 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 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 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- trigger thresholds by only using partial
detector information and delaying the event reconstruction, respectively, have
a reach comparable to the standard 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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