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Register a new userExceptional points of two-dimensional random walks at multiples of the cover time
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We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by $N
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Given a sequence of lattice approximations $D_Nsubsetmathbb Z^2$ of a bounded continuum domain $Dsubsetmathbb R^2$ with the vertices outside $D_N$ fused together into one boundary vertex $varrho$, we consider discrete-time simple random walks in $D_Ncup{varrho}$ run for a time proportional to the expected cover time and describe the scaling limit of the exceptional level sets of the thick, thin, light and avoided points. We show that these are distributed, up a spatially-dependent log-normal factor, as the zero-average Liouville Quantum Gravity measures in $D$. The limit law of the local time configuration at, and nearby, the exceptional points is determined as well. The results extend earlier work by the first two authors who analyzed the continuous-time problem in the parametrization by the local time at $varrho$. A novel uniqueness result concerning divisible random measures and, in particular, Gaussian Multiplicative Chaos, is derived as part of the proofs.
We study continuous-time (variable speed) random walks in random environments on $mathbb{Z}^d$, $dge2$, where, at time $t$, the walk at $x$ jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary and ergodic with respect to space-time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer-Sjostrand representation of gradient models with certain non-strictly convex potentials.
We study a particular class of complex-valued random variables and their associated random walks: the complex obtuse random variables. They are the generalization to the complex case of the real-valued obtuse random variables which were introduced in cite{A-E} in order to understand the structure of normal martingales in $RR^n$.The extension to the complex case is mainly motivated by considerations from Quantum Statistical Mechanics, in particular for the seek of a characterization of those quantum baths acting as classical noises. The extension of obtuse random variables to the complex case is far from obvious and hides very interesting algebraical structures. We show that complex obtuse random variables are characterized by a 3-tensor which admits certain symmetries which we show to be the exact 3-tensor analogue of the normal character for 2-tensors (i.e. matrices), that is, a necessary and sufficient condition for being diagonalizable in some orthonormal basis. We discuss the passage to the continuous-time limit for these random walks and show that they converge in distribution to normal martingales in $CC^N$. We show that the 3-tensor associated to these normal martingales encodes their behavior, in particular the diagonalization directions of the 3-tensor indicate the directions of the space where the martingale behaves like a diffusion and those where it behaves like a Poisson process. We finally prove the convergence, in the continuous-time limit, of the corresponding multiplication operators on the canonical Fock space, with an explicit expression in terms of the associated 3-tensor again.
We introduce a Gibbs measure on nearest-neighbour paths of length $t$ in the Euclidean $d$-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature $beta$. We prove that, for all $beta>0$, the random walk condensates to a set of diameter $(t/beta)^{1/3}$ in dimension $d=2$, up to a multiplicative constant. In all dimensions $dge 3$, we also prove that the volume is bounded above by $(t/beta)^{d/(d+1)}$ and the diameter is bounded below by $(t/beta)^{1/(d+1)}$. Similar results hold for a random walk conditioned to have local time greater than $beta$ everywhere in its range when $beta$ is larger than some explicit constant, which in dimension two is the logarithm of the connective constant.
We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of valleys of height $log t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.
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