Frank den Hollander

Branching random walk in random environment: phase transitions for local and global growth rates

SummaryLet (ηn) be the infinite particle system on ℤ whose evolution is as follows. At each unit of time each particle independently is replaced by a new generation. The size of a new generation descending from a particle at sitex has distributionFx and each of its members independently jumps to sitex±1 with probability (1±h)/2,h∈[0, 1]. The sequence {Fx} is i.i.d. with uniformly bounded second moment and is kept fixed during the evolution. The initial configurationη0 is shift invariant and ergodic.Two quantities are considered: (1)the global particle densityDn (=large volume limit of number of particles per site at timen);(2)the local particle densitydn> (=average number of particles at site 0 at timen).We calculate the limits ϱ and λ ofn−1 log(Dn) andn−1 log(dn) explicitly in the form of two variational formulas. Both limits (and variational formulas) do not depend on the realization of {Fx} a.s. By analyzing the variational formulas we extract how ϱ and λ depend on the drifth for fixed distribution ofFx. It turns out that the system behaves in a way that is drastically different from what happens in a spatially homogeneous medium:(i)Both \g9(h) and \gl(h) exhibit a phase transition associated with localization vs. delocalization at two respective critical valuesh1 andh3 in (0,1). Here the behavior of the path of descent of a typical particle in the whole population resp. in the population at 0 changes from moving on scaleo(n) to moving on scalen. We extract variational expressions forh1 andh3.(ii)Both \g9(h) and \gl(h) change sign at two respective critical valuesh2 andh4 in (0,1) (for suitable distribution ofFx. That is, the system changes from survival to extinction on a global resp. on a local scale.(iii)\g9(h)\>=\gl(h) for allh; \g9(h)=\gl(h) forh sufficiently small and \g9(h)\s>\gl(h) forh sufficiently large. This means that the system develops a clustering phenomenon ash increases: the population has large peaks on a thin set.(iv)\g9(h)\s>0\s>\gl(h) for a range ofh. (extreme clustering of the system)We formulate certain technical properties of the variational formulas that are needed in order to derive the qualitative picture of the phase diagram in its full glory. The proof of these properties is deferred to a forthcoming paper dealing exclusively with functional analytic aspects.The variational formulas reveal a selection mechanism: the typical particle has a path of descent that is best adapted to the given {Fx} and that is atypical under the law of the underlying random walk. The random medium induces “selection of the fittest”.

Invasion percolation on regular trees

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its r-point function for any r ≥ 2 and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.

A new inductive approach to the lace expansion for self-avoiding walks

Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ℤd where loops of length m are penalised by a factor e−β/m p (04, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0.

Methods of contemporary mathematical statistical physics

We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetries and inequalities generally valid for nonequilibrium systems, including the fluctuation symmetry, Jarzynski equality, and the direction of currents. In Part II we concentrate on the macroscopic state and how entropy provides a bridge between microscopic dynamics and macroscopic irreversibility; included is a construction of quantum macroscopic states and a result on the equivalence of ensembles.Reflection Positivity and Phase Transitions in Lattice Spin Models.- Stochastic Geometry of Classical and Quantum Ising Models.- Localization Transition in Disordered Pinning Models.- Metastability.- Three Lectures on Metastability Under Stochastic Dynamics.- A Selection of Nonequilibrium Issues.- Facilitated Spin Models: Recent and New Results.Reflection positivity (RP) is a property of Gibbs measures exhibited by a class of lattice spin systems that include the Ising, Potts and Heisenberg models. The RP property is useful because of its two basic consequences: infrared bound and chessboard estimates. These are one of basic (and rather efficient) tools for proving phase transitions in many models of physical interest. The notes presented hereby summarize the lectures on reflection positivity and its consequences that the author delivered at the Prague Summer School on Mathematical Statistical Mechanics in September 2006. The text features both the classical material on the subject from the late 1970s as well as some of the more recent developments.

Random walk in random scenery: A survey of some recent results

In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On

Quenched Lyapunov Exponent for the Parabolic Anderson Model in a Dynamic Random Environment

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Three lectures on metastability under stochastic dynamics

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How T-cells use large deviations to recognize foreign antigens.

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Ensemble nonequivalence in random graphs with modular structure

, we are given a random walk with i.i.d. increments and a random scenery with i.i.d.\ components. The walk and the scenery are assumed to be independent. RWRS is the random process where time is indexed by

On the equivalence of certain ergodic properties for Gibbs states

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