Cutoff for the East process

The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-window. The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $\nu$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $\nu$, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen ('82) implies a CLT for the location of the front, yielding the cutoff result. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an $O(1)$-window.Comment: 33 pages, 2 figure

CUTOFF AT THE " ENTROPIC TIME " FOR SPARSE MARKOV CHAINS

International audienceWe study convergence to equilibrium for a large class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix P the mass is essentially concentrated on few entries. Moreover, the random environment is such that rows of P are independent and such that the entries are exchangeable within each row. This includes various models of random walks on sparse random directed graphs. The models are generally non reversible and the equilibrium distribution is itself unknown. In this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium, with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of P. As an application, we consider the case where the rows of P are i.i.d. random vectors in the domain of attraction of a Poisson-Dirichlet law with index α ∈ (0, 1). Our main results are based on a detailed analysis of the weight of the trajectory followed by the walker. This approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon

Functional limit theorems for random regular graphs

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemer\'edi argument for estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and Related Field

Random walk on sparse random digraphs

International audienceA finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure

Lower Bounds on the Time/Memory Tradeoff of Function Inversion

We study time/memory tradeoffs of function inversion: an algorithm, i.e., an inverter, equipped with an $s$-bit advice on a randomly chosen function $f\colon [n] \mapsto [n]$ and using $q$ oracle queries to $f$, tries to invert a randomly chosen output $y$ of $f$, i.e., to find $x\in f^{-1}(y)$. Much progress was done regarding adaptive function inversion - the inverter is allowed to make adaptive oracle queries. Hellman [IEEE transactions on Information Theory \u2780] presented an adaptive inverter that inverts with high probability a random $f$. Fiat and Naor [SICOMP \u2700] proved that for any $s,q$ with $s^3 q = n^3$ (ignoring low-order terms), an $s$-advice, $q$-query variant of Hellman\u27s algorithm inverts a constant fraction of the image points of any function. Yao [STOC \u2790] proved a lower bound of $sq\ge n$ for this problem. Closing the gap between the above lower and upper bounds is a long-standing open question. Very little is known for the non-adaptive variant of the question - the inverter chooses its queries in advance. The only known upper bounds, i.e., inverters, are the trivial ones (with $s+q= n$), and the only lower bound is the above bound of Yao. In a recent work, Corrigan-Gibbs and Kogan [TCC \u2719] partially justified the difficulty of finding lower bounds on non-adaptive inverters, showing that a lower bound on the time/memory tradeoff of non-adaptive inverters implies a lower bound on low-depth Boolean circuits. Bounds that, for a strong enough choice of parameters, are notoriously hard to prove. We make progress on the above intriguing question, both for the adaptive and the non-adaptive case, proving the following lower bounds on restricted families of inverters: - Linear-advice (adaptive inverter): If the advice string is a linear function of $f$ (e.g., $A\times f$, for some matrix $A$, viewing $f$ as a vector in $[n]^n$), then $s+q \in \Omega(n)$. The bound generalizes to the case where the advice string of $f_1 + f_2$, i.e., the coordinate-wise addition of the truth tables of $f_1$ and $f_2$, can be computed from the description of $f_1$ and $f_2$ by a low communication protocol. - Affine non-adaptive decoders: If the non-adaptive inverter has an affine decoder - it outputs a linear function, determined by the advice string and the element to invert, of the query answers - then $s \in \Omega(n)$ (regardless of $q$). - Affine non-adaptive decision trees: If the non-adaptive inversion algorithm is a $d$-depth affine decision tree - it outputs the evaluation of a decision tree whose nodes compute a linear function of the answers to the queries - and $q 0$, then $s\in \Omega(n/d \log n)$

The ProPrems trial: investigating the effects of probiotics on late onset sepsis in very preterm infants

BACKGROUND: Late onset sepsis is a frequent complication of prematurity associated with increased mortality and morbidity. The commensal bacteria of the gastrointestinal tract play a key role in the development of healthy immune responses. Healthy term infants acquire these commensal organisms rapidly after birth. However, colonisation in preterm infants is adversely affected by delivery mode, antibiotic treatment and the intensive care environment. Altered microbiota composition may lead to increased colonisation with pathogenic bacteria, poor immune development and susceptibility to sepsis in the preterm infant.Probiotics are live microorganisms, which when administered in adequate amounts confer health benefits on the host. Amongst numerous bacteriocidal and nutritional roles, they may also favourably modulate host immune responses in local and remote tissues. Meta-analyses of probiotic supplementation in preterm infants report a reduction in mortality and necrotising enterocolitis. Studies with sepsis as an outcome have reported mixed results to date.Allergic diseases are increasing in incidence in "westernised" countries. There is evidence that probiotics may reduce the incidence of these diseases by altering the intestinal microbiota to influence immune function. METHODS/DESIGN: This is a multi-centre, randomised, double blinded, placebo controlled trial investigating supplementing preterm infants born at < 32 weeks' gestation weighing < 1500 g, with a probiotic combination (Bifidobacterium infantis, Streptococcus thermophilus and Bifidobacterium lactis). A total of 1,100 subjects are being recruited in Australia and New Zealand. Infants commence the allocated intervention from soon after the start of feeds until discharge home or term corrected age. The primary outcome is the incidence of at least one episode of definite (blood culture positive) late onset sepsis before 40 weeks corrected age or discharge home. Secondary outcomes include: Necrotising enterocolitis, mortality, antibiotic usage, time to establish full enteral feeds, duration of hospital stay, growth measurements at 6 and 12 months' corrected age and evidence of atopic conditions at 12 months' corrected age. DISCUSSION: Results from previous studies on the use of probiotics to prevent diseases in preterm infants are promising. However, a large clinical trial is required to address outstanding issues regarding safety and efficacy in this vulnerable population. This study will address these important issues. TRIAL REGISTRATION: Australia and New Zealand Clinical Trials Register (ANZCTR): ACTRN012607000144415The product "ABC Dophilus Probiotic Powder for Infants®", Solgar, USA has its 3 probiotics strains registered with the Deutsche Sammlung von Mikroorganismen und Zellkulturen (DSMZ--German Collection of Microorganisms and Cell Cultures) as BB-12 15954, B-02 96579, Th-4 15957

Recurrent Immunoglobulin A Nephropathy after Kidney Transplant—An Updated Review

Immunoglobulin A nephropathy (IgAN) is the commonest glomerulonephritis worldwide, a category that represents the third most frequent cause of end-stage kidney disease (ESKD) in the United States. Kidney transplantation remains the optimal treatment of ESKD, and yet the prospects of IgAN recurrence post-transplant dampens the enthusiasm for living kidney donation in some instances, in addition to limiting the longevity of the kidney allograft. Moreover, the lack of a standardized method for detecting IgAN recurrence, since not all centers perform protocol allograft biopsies, has led to an underestimation of the extent of the issue. The pathogenesis of de novo IgAN remains conjectural, let alone the pathways for recurrent disease, but is increasingly recognized as a multi-hit injury mechanism. Identification of recurrent disease rests mainly on clinical symptoms and signs (e.g., hematuria, proteinuria) and could only be definitively proven with histologic evidence which is invasive and prone to sampling error. Treatment had relied mainly on nonspecific goals of proteinuria reduction, and in some cases, immunosuppression for active, crescentic disease. More recently, newer targets have the potential to widen the armamentarium for directed therapies, with more studies on the horizon. This review article provides an update on recurrent IgAN post-transplant