Censored Glauber Dynamics for the mean field Ising Model

We study Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss Model. It is well known that at high temperature ($\beta < 1$) the mixing time is $\Theta(n\log n)$, whereas at low temperature ($\beta > 1$) it is $\exp(\Theta(n))$. Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed $\beta > 1$, the mixing-time of this model is $\Theta(n\log n)$, analogous to the high-temperature regime of the original dynamics. Furthermore, they showed \emph{cutoff} for the original dynamics for fixed $\beta<1$. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Currie-Weiss model. Namely, we found a scaling window of order $1/\sqrt{n}$ around the critical temperature $\beta_c=1$, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if $\beta = 1 + \delta$ for some $\delta > 0$ with $\delta^2 n \to \infty$, then the mixing-time has order $(n / \delta)\log(\delta^2 n)$. The cutoff constant is $(1/2+[2(\zeta^2 \beta / \delta - 1)]^{-1})$, where $\zeta$ is the unique positive root of $g(x)=\tanh(\beta x)-x$, and the cutoff window has order $n / \delta$.Comment: 55 pages, 4 figure

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 for the Ising model on the lattice

Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in $L^1$ on a system of size $n$ is $O(\log n)$. Whether in this regime there is cutoff, i.e. a sharp transition in the $L^1$-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at $(c+o(1))\log n$ for some fixed $c>0$, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For $\Z^2$ this carries all the way to the critical temperature. Specifically, for fixed $d\geq 1$, the continuous-time Glauber dynamics for the Ising model on $(\Z/n\Z)^d$ with periodic boundary conditions has cutoff at $(d/2\lambda_\infty)\log n$, where $\lambda_\infty$ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating $L^1$ to $L^2$ mixing which enables the application of log-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems.Comment: 34 pages, 3 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

Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)

Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure

Mixing time of critical Ising model on trees is polynomial in the height

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature $\beta_c$, the inverse-gap is bounded for $\beta < \beta_c$, polynomial in the surface area for $\beta = \beta_c$ and exponential in it for $\beta > \beta_c$. This has been proved for $\Z^2$ except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for $\beta \beta_c$ were established, where $\beta_c$ is the critical spin-glass parameter, and the tree-height $h$ plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the $b$-ary tree, by showing that it is indeed polynomial in $h$ at criticality. The degree of our polynomial bound does not depend on $b$, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for $\beta > \beta_c$, the inverse-gap and mixing-time are both $\exp[\Theta((\beta-\beta_c) h)]$.Comment: 53 pages; 3 figure

Glauber Dynamics for the mean-field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\geq 3$ states and show that it undergoes a critical slowdown at an inverse-temperature $\beta_s(q)$ strictly lower than the critical $\beta_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $\beta_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $\beta<\beta_s(q)$ the mixing time is asymptotically $C(\beta, q) n \log n$ and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order $n$. At $\beta=\beta_s(q)$ the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order $n^{4/3}$. For $\beta>\beta_s(q)$ the mixing time is exponentially large in $n$. Furthermore, as $\beta \uparrow \beta_s$ with $n$, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of $O(n^{-2/3})$ around $\beta_s$. These results form the first complete analysis of mixing around the critical dynamical temperature --- including the critical power law --- for a model with a first order phase transition.Comment: 45 pages, 5 figure

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