Applications of Stein's method for concentration inequalities

Stein's method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie--Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erd\H{o}s--R\'{e}nyi random graph $G(n,p)$ when $p\ge0.31$. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.Comment: Published in at http://dx.doi.org/10.1214/10-AOP542 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Central limit theorem for first-passage percolation time across thin cylinders

We prove that first-passage percolation times across thin cylinders of the form $[0,n]\times [-h_n,h_n]^{d-1}$ obey Gaussian central limit theorems as long as $h_n$ grows slower than $n^{1/(d+1)}$. It is an open question as to what is the fastest that $h_n$ can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of $(1,0,...,0)$, we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also provide some numerical evidence in support of the conjecture in dimension 2.Comment: Final version, accepted in Probability Theory and Related Fields. 40 pages, 7 figure

Jigsaw percolation: What social networks can collaboratively solve a puzzle?

We introduce a new kind of percolation on finite graphs called jigsaw percolation. This model attempts to capture networks of people who innovate by merging ideas and who solve problems by piecing together solutions. Each person in a social network has a unique piece of a jigsaw puzzle. Acquainted people with compatible puzzle pieces merge their puzzle pieces. More generally, groups of people with merged puzzle pieces merge if the groups know one another and have a pair of compatible puzzle pieces. The social network solves the puzzle if it eventually merges all the puzzle pieces. For an Erd\H{o}s-R\'{e}nyi social network with $n$ vertices and edge probability $p_n$, we define the critical value $p_c(n)$ for a connected puzzle graph to be the $p_n$ for which the chance of solving the puzzle equals $1/2$. We prove that for the $n$-cycle (ring) puzzle, $p_c(n)=\Theta(1/\log n)$, and for an arbitrary connected puzzle graph with bounded maximum degree, $p_c(n)=O(1/\log n)$ and $\omega(1/n^b)$ for any $b>0$. Surprisingly, with probability tending to 1 as the network size increases to infinity, social networks with a power-law degree distribution cannot solve any bounded-degree puzzle. This model suggests a mechanism for recent empirical claims that innovation increases with social density, and it might begin to show what social networks stifle creativity and what networks collectively innovate.Comment: Published at http://dx.doi.org/10.1214/14-AAP1041 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Central Limit Theorem for Gram-Schmidt Random Walk Design

We prove a central limit theorem for the Horvitz-Thompson estimator based on the Gram-Schmidt Walk (GSW) design, recently developed in Harshaw et al.(2022). In particular, we consider the version of the GSW design which uses randomized pivot order, thereby answering an open question raised in the same article. We deduce this under minimal and global assumptions involving only the problem parameters such as the (sum) potential outcome vector and the covariate matrix. As an interesting consequence of our analysis we also obtain the precise limiting variance of the estimator in terms of these parameters which is smaller than the previously known upper bound. The main ingredients are a simplified skeletal process approximating the GSW design and concentration phenomena for random matrices obtained from random sampling using the Stein's method for exchangeable pairs.Comment: 35 pages. Some typo's fixed in the arxiv abstract to fit arxiv's abstract requirement

Affinity binding of non-histone chromatin proteins to the X chromosome of Drosophila by in situ chromatin reconstitution and its significance

Cytophotometric analysis of the in situ binding affinity of non-histone chromosomal protein (NHCP) to the polytenic X chromosome and autosome of Drosophila melanogaster has been carried out using Feulgen-Napthol Yellow S staining technique. The results reveal that the mean transformed absorbance ratio (male:female) with a 547nm interference band filter for the two specific segments of the X chromosome is close to 0-5, while for a specific segment of an autosome it is close to l"0, in the two sets of control; namely, the positive control (no treatment) and the negative control (treated with 1 M-urea+2M-NaCl) as well as in the reconstituted chromosomal preparations, which received 1 M-urea+2M-NaCl and the NHCP isolated from D. melanogaster. In contrast, the transformed absorbance ratios (male:female) with a 433 nm interference band filter yielded an interestingly different result. The ratios with a 433 nm filter for the X chromosome segments are significantly greater than 0-5 in all three sets of experiments. This finding by itself suggests that the NHCP binding affinity is dissimilar for the X chromosomes of male and female. When the 433 to 547 nm absorbance ratios were compared among the three sets, the data clearly revealed that in both positive control and NHCP reconstituted samples, the absorbance ratios (i.e. 433:547 nm) are significantly different between X chromosomes from males and those from females, while they are different between autosomes from males and females. The ratios are also not significantly different between male and female, either for the X chromosome or for the autosome in the negative control. These findings, therefore, suggest that there is a stronger binding affinity of NHCP for the male X chromosome of Drosophila, and reinstate the view that the X chromosomal hyperactivity in male Drosophila is the consequence of a regulated organizational change in the DNA template

Bacteriological examination of drinking water in Burdwan, India with reference to coliforms

Most probable number (MPN) test was done to detect the coliform in water samples collected from mobile vendors, sweet shops and tap water supplied from Burdwan municipality. The study revealed that the number of coliforms was very high (1600) in water samples collected from mobile vendors. The bacteria were identified as Escherichia coli. Bacteriological examination of water samples collected from different sources showed that the water of mobile vendors and sweet shops of Burdwan marketarea was not potable while the municipal tap water was found to be safe for drinking

Ferroelectric order associated with an ordered occupancy at the octahedral site of the inverse spinel structure of multiferroic NiFe2O4

We report a ferroelectric order at ~ 98 K for NiFe2O4, which carries an inverse spinel structure with a centrosymmetric Fd3m structure at room temperature. The value of spontaneous electric polarization is considerably high as ~ 0.29 {\mu}C/cm2 for 5 kV/cm poling field. The electric polarization decreases considerably (~ 17 %) around liquid nitrogen temperature upon application of 50 kOe field, proposing a significant magnetoelectric coupling. The synchrotron diffraction studies confirm a structural transition at ~ 98 K to a noncentrosymmetric structure of P4122 space group. The occurrence of polar order is associated with an ordered occupancy of Ni and Fe atoms at the octahedral sites of the P4122 structure, instead of random occupancies at the octahedral site of the inverse spinel structure. The results propose that NiFe2O4 is a new type-II multiferroic material.Comment: 8 pages, 7 figure