arXiv subjects

Alexander Drewitz

Publications and source records attributed to Alexander Drewitz.

At least 37 records · Page 2Linked to original sources

arXiv2021

(Un-)bounded transition fronts for the parabolic Anderson model and the randomized F-KPP equation

We investigate the uniform boundedness of the fronts of the solutions to the randomized Fisher-KPP equation and to its linearization, the parabolic Anderson model. It has been known that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the special case of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one has a uniformly bounded (in time) transition front. Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. Nevertheless, we establish that this property does hold true for the parabolic Anderson model.

math.AP↗

arXiv2021

Invariance principles and Log-distance of F-KPP fronts in a random medium

We study the front of the solution to the F-KPP equation with randomized non-linearity. Under suitable assumptions on the randomness involving spatial mixing behavior and boundedness, we show that the front of the solution lags at most logarithmically in time behind the front of the solution of the corresponding linearized equation, i.e. the parabolic Anderson model. This can be interpreted as a partial generalization of Bramson's findings for the homogeneous setting. Building on this result, we establish functional central limit theorems for the fronts of the solutions to both equations.

math.PR↗

arXiv2019

Random Walk Among Mobile/Immobile Traps: A Short Review

There have been extensive studies of a random walk among a field of immobile traps (or obstacles), where one is interested in the probability of survival as well as the law of the random walk conditioned on its survival up to time $t$. In contrast, very little is known when the traps are mobile. We will briefly review the literature on the trapping problem with immobile traps, and then review some recent results on a model with mobile traps, where the traps are represented by a Poisson system of independent random walks on ${\mathbb Z}^d$. Some open questions will be given at the end.

math.PR↗

arXiv2019

Quenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson model

We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher-KPP equation with random branching and its linearized version, the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher-KPP equation fulfill quenched invariance principles. In addition, we prove that at time t the distance between the median of the maximal particle of the BRWRE and the front of the solution of the PAM is in O(ln t). This partially transfers results from Bramson [Comm. Pure Appl. Math. 31 (1978), no. 5, 531--581] to the setting of BRWRE.

math.PR↗

arXiv2018

The sign clusters of the massless Gaussian free field percolate on $\mathbb{Z}^d$, $d \geqslant 3$ (and more)

We investigate the percolation phase transition for level sets of the Gaussian free field on $\mathbb{Z}^d$, with $d\geqslant 3$, and prove that the corresponding critical parameter $h_*(d)$ is strictly positive for all $d\geqslant3$, thus settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on $\mathbb{Z}^d$, for all $d\geqslant 3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of $\mathbb{Z}^d$ with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over $\mathbb{Z}^d$.

math.PR↗

arXiv2016

Subdiffusivity of a random walk among a Poisson system of moving traps on ${\mathbb Z}$

We consider a random walk among a Poisson system of moving traps on ${\mathbb Z}$. In earlier work [DGRS12], the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time $t$ in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range.

math.PR↗

arXiv2015

High-dimensional asymptotics for percolation of Gaussian free field level sets

We consider the Gaussian free field on $\mathbb{Z}^d$, $d$ greater or equal to $3$, and prove that the critical density for percolation of its level sets behaves like $1/d^{1 + o(1)}$ as $d$ tends to infinity. Our proof gives the principal asymptotic behavior of the corresponding critical level $h_*(d)$. Moreover, it shows that a related parameter $h_{**}(d) \geq h_*(d)$ introduced by Rodriguez and Sznitman in arXiv:1202.5172 is in fact asymptotically equivalent to $h_*(d)$.

math.PR↗

arXiv2015

Asymptotics for Lipschitz percolation above tilted planes

We consider Lipschitz percolation in $d+1$ dimensions above planes tilted by an angle $γ$ along one or several coordinate axes. In particular, we are interested in the asymptotics of the critical probability as $d \to \infty$ as well as $γ\to π/4.$ Our principal results show that the convergence of the critical probability to 1 is polynomial as $d\to \infty$ and $γ\to π/4.$ In addition, we identify the correct order of this polynomial convergence and in $d=1$ we also obtain the correct prefactor.

math.PR↗

arXiv2014

On chemical distances and shape theorems in percolation models with long-range correlations

In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. We also obtain alternative proofs to the main results in arXiv:1111.3979. Finally, as a corollary, we obtain new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.

math.PR↗

arXiv2013

Transience of the vacant set for near-critical random interlacements in high dimensions

The model of random interlacements is a one-parameter family $\mathcal I^u,$ $u \ge 0,$ of random subsets of $\mathbb{Z}^d,$ which locally describes the trace of simple random walk on a $d$-dimensional torus run up to time $u$ times its volume. Its complement, the so-called vacant set $\mathcal V^u$, has been shown to undergo a non-trivial percolation phase-transition in $u;$ i.e., there exists $u_*(d) \in (0, \infty)$ such that for $u \in [0, u_*(d))$ the vacant set $\mathcal V^u$ contains a unique infinite connected component $\mathcal V_\infty^u,$ while for $u > u_*(d)$ it consists of finite connected components. Sznitman \cite{SZ11,SZ11B} showed that $u_*(d) \sim \log d,$ and in this article we show the existence of $u(d) > 0$ with $\frac{u(d)}{u_*(d)} \to 1$ as $d \to \infty$ such that $\mathcal V_\infty^{u}$ is transient for all $u \in [0, u(d)).$

math.PR↗

arXiv2013

Local percolative properties of the vacant set of random interlacements with small intensity

Random interlacements at level u is a one parameter family of connected random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the vacant set at level u, exhibits a non-trivial percolation phase transition in u, as shown in arXiv:0704.2560 and arXiv:0808.3344, and the infinite connected component, when it exists, is almost surely unique, see arXiv:0805.4106. In this paper we study local percolative properties of the vacant set of random interlacements at level u for all dimensions d>=3 and small intensity parameter u>0. We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at level u. Our results imply that finite connected components of the vacant set at level u are unlikely to be large. These results were proved in arXiv:1002.4995 for d>=5. Our approach is different from that of arXiv:1002.4995 and works for all d>=3. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.

math.PR↗

arXiv2013

Selected Topics in Random Walk in Random Environment

Random walk in random environment (RWRE) is a fundamental model of statistical mechanics, describing the movement of a particle in a highly disordered and inhomogeneous medium as a random walk with random jump probabilities. It has been introduced in a series of papers by Chernov and Temkin as a model for DNA chain replication and crystal growth, and also as a model for turbulent behavior in fluids through a Lorentz gas description by Sinai. It is a simple but powerful model for a variety of complex large-scale disordered phenomena arising from fields such as physics, biology and engineering. While the one-dimensional model is well-understood, in the multidimensional setting, fundamental questions about the RWRE model have resisted repeated and persistent attempts to answer them. Two major complications in this context stem from the loss of the Markov property under the averaged measure as well as the fact that in dimensions larger than one, the RWRE is not reversible anymore. In these notes we present a general overview of the model, with an emphasis on the multidimensional setting and a more detailed description of recent progress around ballisticity questions.

math.PR↗

arXiv2013

Effective Polynomial Ballisticity Condition for Random Walk in Random Environment

The conditions $(T)_γ,$ $γ\in (0,1),$ which have been introduced by Sznitman in 2002, have had a significant impact on research in random walk in random environment. Among others, these conditions entail a ballistic behaviour as well as an invariance principle. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. The main goal of this paper is to show that in all relevant dimensions (i.e., $d \ge 2$), in order to establish the conditions $(T)_γ$, it is actually enough to check a corresponding condition $(\mathcal{P})$ of polynomial type. In addition to only requiring an a priori weaker decay of the corresponding slab exit probabilities than $(T)_γ,$ another advantage of the condition $(\mathcal{P})$ is that it is effective in the sense that it can be checked on finite boxes. In particular, this extends the conjectured equivalence of the conditions $(T)_γ,$ $γ\in (0,1),$ to all relevant dimensions.

math.PR↗

arXiv2013

Symmetric Rearrangements Around Infinity with Applications to Levy Processes

We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (1976) and can be interpreted as involving symmetric rearrangements of domains around infinity. As applications, we prove two comparison results for general Levy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Levy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Levy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi (2011) for the Wiener sausage. In the second application, we show that the q-capacity of a Borel measurable set for a Levy process can only decrease if the set and the Levy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (1983) for symmetric Levy processes.

math.PR↗

arXiv2012

Level 1 quenched large deviation principle for random walk in dynamic random environment

Consider a random walk in a time-dependent random environment on the lattice Zd. Recently, Rassoul-Agha, Seppalainen and Yilmaz [RSY11] proved a general large deviation principle under mild ergodicity assumptions on the random environment for such a random walk, establishing first level 2 and 3 large deviation principles. Here we present two alternative short proofs of the level 1 large deviations under mild ergodicity assumptions on the environment: one for the continuous time case and another one for the discrete time case. Both proofs provide the existence, continuity and convexity of the rate function. Our methods are based on the use of the sub-additive ergodic theorem as presented by Varadhan in 2003.

math.PR↗

arXiv2012

Survival Probability of a Random Walk Among a Poisson System of Moving Traps

We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on Z^d, which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e^{-λ_1\sqrt{t}} for d=1, as e^{-λ_2 t/\log t} for d=2, and as e^{-λ_d t} for d>= 3, where λ_1 and λ_2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e^{-\tilde λ_d t}, with \tilde λ_d>0 for all d>= 1. A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous time symmetric random walk increases under perturbation by a deterministic path.

math.PR↗

arXiv2012

Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment

Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each $γ\in(0,1)$ the ballisticity condition $(T)_γ$ and the condition $(T')$ defined as the fulfillment of $(T)_γ$ for each $γ\in(0,1)$. Sznitman proved that $(T')$ implies a ballistic law of large numbers. Furthermore, he showed that for all $γ\in (0.5,1)$, $(T)_γ$ is equivalent to $(T')$. Recently, Berger has proved that in dimensions larger than three, for each $γ\in (0,1)$, condition $(T)_γ$ implies a ballistic law of large numbers. On the other hand, Drewitz and Ram\'{ı}rez have shown that in dimensions $d\ge2$ there is a constant $γ_d\in(0.366,0.388)$ such that for each $γ\in(γ_d,1)$, condition $(T)_γ$ is equivalent to $(T')$. Here, for dimensions larger than three, we extend the previous range of equivalence to all $γ\in(0,1)$. For the proof, the so-called effective criterion of Sznitman is established employing a sharp estimate for the probability of atypical quenched exit distributions of the walk leaving certain boxes. In this context, we also obtain an affirmative answer to a conjecture raised by Sznitman in 2004 concerning these probabilities. A key ingredient for our estimates is the multiscale method developed recently by Berger.

math.PR↗

arXiv2009

Ballisticity conditions for random walk in random environment

Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions $d\ge 2$. In 2002, Sznitman introduced for each $γ\in (0,1)$ the ballisticity conditions $(T)_γ$ and $(T'),$ the latter being defined as the fulfilment of $(T)_γ$ for all $γ\in (0,1).$ He proved that $(T')$ implies ballisticity and that for each $γ\in (0.5,1),$ $(T)_γ$ is equivalent to $(T')$. It is conjectured that this equivalence holds for all $γ\in (0,1).$ Here we prove that for $γ\in (γ_d,1),$ where $γ_d$ is a dimension dependent constant taking values in the interval $(0.366,0.388),$ $(T)_γ$ is equivalent to $(T').$ This is achieved by a detour along the effective criterion, the fulfilment of which we establish by a combination of techniques developed by Sznitman giving a control on the occurrence of atypical quenched exit distributions through boxes.

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