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Register a new userSelf-intersection local times of random walks: Exponential moments in subcritical dimensions
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Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $Z^d$ up to time $t$. This is the $p$-norm of the vector of the walkers local times, $ell_t$. We derive precise logarithmic asymptotics of the expectation of $exp{theta_t |ell_t|_p}$ for scales $theta_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $|ell_t|_p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[|ell_t|_p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^{1/d}$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.
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We prove that the local time process of a planar simple random walk, when time is scaled logarithmically, converges to a non-degenerate pure jump process. The convergence takes place in the Skorokhod space with respect to the $M1$ topology and fails to hold in the $J1$ topology.
In the parking model on $mathbb{Z}^d$, each vertex is initially occupied by a car (with probability $p$) or by a vacant parking spot (with probability $1-p$). Cars perform independent random walks and when they enter a vacant spot, they park there, thereby rendering the spot occupied. Cars visiting occupied spots simply keep driving (continuing their random walk). It is known that $p=1/2$ is a critical value in the sense that the origin is a.s. visited by finitely many distinct cars when $p<1/2$, and by infinitely many distinct cars when $pgeq 1/2$. Furthermore, any given car a.s. eventually parks for $p leq 1/2$ and with positive probability does not park for $p > 1/2$. We study the subcritical phase and prove that the tail of the parking time $tau$ of the car initially at the origin obeys the bounds [ expleft( - C_1 t^{frac{d}{d+2}}right) leq mathbb{P}_p(tau > t) leq expleft( - c_2 t^{frac{d}{d+2}}right) ] for $p>0$ sufficiently small. For $d=1$, we prove these inequalities for all $p in [0,1/2)$. This result presents an asymmetry with the supercritical phase ($p>1/2$), where methods of Bramson--Lebowitz imply that for $d=1$ the corresponding tail of the parking time of the parking spot of the origin decays like $e^{-csqrt{t}}$. Our exponent $d/(d+2)$ also differs from those previously obtained in the case of moving obstacles.
We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].
Let $B^{alpha_i}$ be an $(N_i,d)$-fractional Brownian motion with Hurst index ${alpha_i}$ ($i=1,2$), and let $B^{alpha_1}$ and $B^{alpha_2}$ be independent. We prove that, if $frac{N_1}{alpha_1}+frac{N_2}{alpha_2}>d$, then the intersection local times of $B^{alpha_1}$ and $B^{alpha_2}$ exist, and have a continuous version. We also establish H{o}lder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points. One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre ({it J. Theor. Probab.} {bf 20} (2007)), where the existence of the intersection local times of two independent $(1,d)$-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.
Consider an arbitrary transient random walk on $Z^d$ with $dinN$. Pick $alphain[0,infty)$ and let $L_n(alpha)$ be the spatial sum of the $alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $alpha$, $L_n(alpha)$ is the number of the $alpha$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(alpha)$ as $ntoinfty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by v{C}erny cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.
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