Showing papers by "Amir Dembo published in 2017"

Extremal cuts of sparse random graphs

Abstract: For Erdős–Renyi random graphs with average degree γγ, and uniformly random γγ-regular graph on nn vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are n(γ4+P∗γ4−−√+o(γ−−√))+o(n)n(γ4+P∗γ4+o(γ))+o(n) while the size of the minimum bisection is n(γ4−P∗γ4−−√+o(γ−−√))+o(n)n(γ4−P∗γ4+o(γ))+o(n). Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington–Kirkpatrick model, with P∗≈0.7632P∗≈0.7632 standing for the ground state energy of the latter, expressed analytically via Parisi’s formula.

Persistence of Gaussian processes: non-summable correlations

Abstract: Suppose the auto-correlations of real-valued, centered Gaussian process $$Z(\cdot )$$ are non-negative and decay as $$\rho (|s-t|)$$ for some $$\rho (\cdot )$$ regularly varying at infinity of order $$-\alpha \in [-1,0)$$ . With $$I_\rho (t)=\int _0^t \rho (s)ds$$ its primitive, we show that the persistence probabilities decay rate of $$ -\log \mathbb {P}(\sup _{t \in [0,T]}\{Z(t)\}<0)$$ is precisely of order $$(T/I_\rho (T)) \log I_\rho (T)$$ , thereby closing the gap between the lower and upper bounds of Newell and Rosenblatt (Ann. Math. Stat. 33:1306–1313, 1962), which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of Sakagawa (Adv. Appl. Probab. 47:146–163, 2015) about the dependence on d of such persistence decay for the Langevin dynamics of certain $$ abla \phi $$ -interface models on $$\mathbb {Z}^d$$ .

The Criticality of a Randomly-Driven Front

Abstract: Consider an advancing `front' $ R(t) \in \mathbb{Z}_{\geq 0} $ and particles performing independent continuous time random walks on $ (R(t),\infty)\cap\mathbb{Z} $ Starting at $R(0)=0$, whenever a particle attempts to jump into $R(t)$ the latter instantaneously moves $k \ge 1$ steps to the right, absorbing all particles along its path We take $ k $ to be the minimal random integer such that exactly $ k $ particles are absorbed by the move of $ R $, and view the particle system as a discrete version of the Stefan problem \begin{align*} &\partial_t u_*(t,\xi) = \tfrac12 \partial^2_{\xi} u_*(t,\xi), \quad \xi >r(t), &u_*(t,r(t))=0, &\tfrac{d~}{dt}r(t) = \tfrac12 \partial_\xi u_*(t,r(t)), &t\mapsto r(t) \text{ non-decreasing }, \quad r(0):=0 \end{align*} For a constant initial particles density $u_*(0,\xi)=\rho {\bf 1}_{\{\xi >0\}}$, at $\rho<1$ the particle system and the PDE exhibit the same diffusive behavior at large time, whereasat $\rho \ge 1$ the PDE explodes instantaneously Focusing on the critical density $ \rho=1 $, we analyze the large time behavior of the front $ R(t) $ for the particle system, and obtain both the scaling exponent of $R(t)$ and an explicit description of its random scaling limit Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is sensitive to the amount of initial local fluctuations Further, the scaling limit demonstrates an interesting oscillation between instantaneous super- and sub-critical phases Our method is based on a novel monotonicity as well as PDE-type estimates

Equilibrium fluctuation of the Atlas model

Abstract: We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite (Z+Z+-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on R+R+. In this context, we show that the joint law of ranked particles, after being centered and scaled by t−14t−14, converges as t→∞t→∞ to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on R+R+ with the Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional Brownian Motion (fBM). In particular, we prove a conjecture of Pal and Pitman [Ann. Appl. Probab. 18 (2008) 2179–2207] about the asymptotic Gaussian fluctuation of the ranked particles.

Transience in growing subgraphs via evolving sets

Abstract: Nous generalisons la methode basee sur l’evolution aleatoire d’ensembles au cas de modeles de conductances variant avec le temps. Nous l’utilisons pour prouver des bornes superieures sur le noyau de la chaleur. Ceci montre la transitivite de n’importe quelle marche aleatoire faineante, dans $\mathbb{Z}^{d}$, $d\ge3$, avec des conductances par aretes (bornees uniformement superieurement et inferieurement) variant independamment en temps en fonction des conductances par sites. Ceci repond partiellement a la Conjecture 7.1 (Random walk in changing environment (2015) Preprint).

The infinite Atlas process: Convergence to equilibrium.

Abstract: The semi-infinite Atlas process is a one-dimensional system of Brownian particles, where only the leftmost particle gets a unit drift to the right. Its particle spacing process has infinitely many stationary measures, with one distinguished translation invariant reversible measure. We show that the latter is attractive for a large class of initial configurations of slowly growing (or bounded) particle densities. Key to our proof is a new estimate on the rate of convergence to equilibrium for the particle spacing in a triangular array of finite, large size systems.

Brownian Particles with Rank-Dependent Drifts: Out-of-Equilibrium Behavior

Abstract: We study the long-range asymptotic behavior for an out-of-equilibrium countable one-dimensional system of Brownian particles interacting through their rank-dependent drifts. Focusing on the semi-infinite case, where only the leftmost particle gets a constant drift to the right, we derive and solve the corresponding one- sided Stefan (free-boundary) equations. Via this solution we explicitly determine the limiting particle-density profile as well as the asymptotic trajectory of the leftmost particle. While doing so we further establish stochastic domination and convergence to equilibrium results for the vector of relative spacings among the leading particles.

Random walks among time increasing conductances: heat kernel estimates

Abstract: For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved.

Cutoff for lamplighter chains on fractals

Abstract: We show that the total-variation mixing time of the lamplighter random walk on fractal graphs exhibit sharp cutoff when the underlying graph is transient (namely of spectral dimension greater than two). In contrast, we show that such cutoff can not occur for strongly recurrent underlying graphs (i.e. of spectral dimension less than two).

Ferromagnetic Ising measures on large locally tree-like graphs

Abstract: We consider the ferromagnetic Ising model on a sequence of graphs GnGn converging locally weakly to a rooted random tree. Generalizing [Probab. Theory Related Fields 152 (2012) 31–51], under an appropriate “continuity” property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with ++ and −− boundary conditions on that tree. Under the extra assumptions that GnGn are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization is the Ising measure with ++ boundary condition on the limiting tree. The “continuity” property holds except possibly for countable many choices of ββ, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton–Watson trees.

On the geometry of chemical reaction networks: Lyapunov function and large deviations

Abstract: In an earlier paper, we proved the validity of large deviations theory for the particle approximation of quite general chemical reaction networks (CRNs). In this paper, we extend its scope and present a more geometric insight into the mechanism of that proof, exploiting the notion of spherical image of the reaction polytope. This allows to view the asymptotic behavior of the vector field describing the mass-action dynamics of chemical reactions as the result of an interaction between the faces of this polytope in different dimensions. We also illustrate some local aspects of the problem in a discussion of Wentzell-Freidlin (WF) theory, together with some examples.

Universal Persistence for Local Time of One-dimensional Random Walk

Abstract: We prove the power law decay $p(t,x) \sim t^{-\phi(x,b)/2}$ in which $p(t,x)$ is the probability that the fraction of time up to $t$ in which a random walk $S$ of i.i.d. zero-mean increments taking finitely many values, is non-negative, exceeds $x$ throughout $s \in [1,t]$. Here $\phi(x,b)= \mathbb{P}(\text{Levy}(1/2,\kappa(x,b))<0)$ for $\kappa(x,b) = \frac{\sqrt{1-x} b - \sqrt{1+x}}{\sqrt{1-x} b + \sqrt{1+x}}$ and $b=b_S \geq 0$ measuring the asymptotic asymmetry between positive and negative excursions of the walk (with $b_s=1$ for symmetric increments).

Large deviations theory for Markov jump models of chemical reaction networks

Abstract: We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it we further establish the corresponding Wentzell-Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemical reaction networks under mass action kinetics, on a microscopic scale. We provide natural sufficient topological conditions for the applicability of our LDP and W-F results. This then justifies the computation of non-equilibrium potential and exponential transition time estimates between different attractors in the large volume limit, for systems that are beyond the reach of standard chemical reaction network theory.

Exponential concentration for zeroes of stationary Gaussian processes

Abstract: We show that for any centered stationary Gaussian process of integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$ is within $\eta T$ of its mean value, up to an exponentially small in $T$ probability.