Let (Z_n)_{n\in\N_0} be a d-dimensional random walk in random scenery, i.e., Z_n=\sum_{k=0}^{n-1}Y_{S_k} with (S_k)_{k\in\N_0} a random walk in Z^d and (Y_z)_{z\in Z^d} an i.i.d. scenery, independent of the walk.
We assume that the random variables Y_z have a stretched exponential tail. In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of Pr(Z_n>t_n n) for all sequences (t_n)_{n\in\N} satisfying a certain lower bound. This complements previous results, where it was assumed that Y_z has exponential moments of all orders. In contrast to the previous situation,the event \{Z_n>t_nn\} is not realized by a homogeneous behavior of the walk's local times and the scenery, but by many visits of the walker to a particular site and a large value of the scenery at that site. This reflects a well-known extreme behavior typical for random variables having no exponential moments.
We consider Random Walk in Random Scenery, denoted $X_n$, where the random walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of i.i.d random variables with a stretched exponential tail decay, with exponent $α$ with $1<α$. We present asymptotics for the probability, over both randomness, that $\{X_n>n^β\}$ for $1/2<β<1$. To obtain such asymptotics, we establish large deviations estimates for the the self-intersection local times process.
We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time aymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.
A {\it scenery} is a coloring $ξ$ of the integers. Let $\{S_t\}_{t\geq 0}$ be a recurrent random walk on the integers. Observing the scenery $ξ$ along the path of this random walk, one sees the color $χ_t:=ξ(S_t)$ at time $t$. The {\it scenery reconstruction problem} is concerned with recovering the scenery $ξ$, given only the sequence of observations $χ:=(χ_t)_{t\geq 0}$. The scenery reconstruction methods presented to date require the random walk to have bounded increments. Here, we present a new approach for random walks with unbounded increments which works when the tail of the increment d…
This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102--128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.
We consider Sinai's walk in i.i.d. random scenery and focus our attention on a conjecture of Révész \cite{r05} concerning the upper limits of Sinai's walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai's walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has "thin" negative tails and is false otherwise.
We consider a d-dimensional random walk in random scenery X(n), where the scenery consists of i.i.d. with exponential moments but a tail decay of the form exp(-c t^a) with a<d/2. We study the probability, when averaged over both randomness, that {X(n)>ny}. We show that this probability is of order exp(-(ny)^b) with b=a/(a+1).
It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ ξ_j $ for $ j \in \mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $ j.$ The random reward (or scenery) $ ξ_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion …
Let $(Z_n)_{n\in\N}$ be a $d$-dimensional {\it random walk in random scenery}, i.e., $Z_n=\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in\N_0}$ a random walk in $\Z^d$ and $(Y(z))_{z\in\Z^d}$ an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and finite variance. We identify the speed and the rate of the logarithmic decay of $¶(\frac 1n Z_n>b_n)$ for various choices of sequences $(b_n)_n$ in $[1,\infty)$. Depending on $(b_n)_n$ and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions.…
The presence of frozen-in or quenched disorder in a system can often modify the nature of its phase transition. A particular instance of this phenomenon is the so-called rounding effect: it has been shown in many cases that the free-energy curve of the disordered system at its critical point is smoother than that of the homogenous one. In particular some disordered systems do not allow first-order transitions. We study this phenomenon for the pinning of a renewal with stretched-exponential tails on a defect line (the distribution $K$ of the renewal increments satisfies $K(n) \sim c_K\exp(-n^α)…
We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X \geq t] \sim a \exp(-λt^r)$ for some constants $a, λ> 0$ where $r \in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $ r \in (0, 2/3)$ and $ r \in (…
We study the large deviation probabilities of infinite weighted sums of independent random variables that have stretched exponential tails. This generalizes Kiesel and Stadtmüller (2000), who study the same objects under the assumption of finite exponential moments, and Gantert et al.\ (2014), who study finite weighted sums with stretched exponential tails.
We prove large deviation results for the position of the rightmost particle, denoted by $M_n$, in a one-dimensional branching random walk in a case when Cramér's condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e.~both tails decay as $e^{-|t|^r}$ for some $r\in( 0,1)$. It is known that in this case, $M_n$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_n$ . In the proof we use a comparison with the maximum of (a random num…
We consider the probability that a weighted sum of $n$ i.i.d. random variables $X_j$, $j = 1, . . ., n$, with stretched exponential tails is larger than its expectation and determine the rate of its decay, under suitable conditions on the weights. We show that the decay is subexponential, and identify the rate function in terms of the tails of $X_j$ and the weights. Our result generalizes the large deviation principle given by Kiesel and Stadtmüller [8] as well as the tail asymptotics for sums of i.i.d. random variables provided by Nagaev [10, 11]. As an application of our result, motivated by…
We consider Glauber dynamics reversible with respect to Gibbs measures with heavy tails. Spins are unbounded. The interactions are bounded and finite range. The self potential enters into two classes of measures, $κ$-concave probability measure and sub-exponential laws, for which it is known that no exponential decay can occur. We prove, using coercive inequalities, that the associated infinite volume semi-group decay to equilibrium polynomially and stretched exponentially, respectively. Thus improving and extending previous results by Bobkov and Zegarlinski.
We study the parabolic Anderson problem, that is, the heat equation $\partial_tu=Δu+ξu$ on $(0,\infty)\times{\mathbb{Z}}^d$ with independent identically distributed random potential $\{ξ(z):z\in{\mathbb{Z}}^d\}$ and localized initial condition $u(0,x)={\mathbf{1}}_0(x)$. Our interest is in the long-term behavior of the random total mass $U(t)=\sum_zu(t,z)$ of the unique nonnegative solution in the case that the distribution of $ξ(0)$ is heavy tailed. For this, we study two paradigm cases of distributions with infinite moment generating functions: the case of polynomial or Pareto tails, and the…
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 $p\geq 1/2$. Furthermore, any given car a.s. eventually…
In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${\rm T}$. Our main results prove that for any $ξ>0$ and $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$ decays as $\exp{(-(2dξ+o(1))n)}$ with a time constant $μ$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.
We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindelöf integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by S. V. Nagaev (1973). The theorems generalize five theorems by A. V. Nagaev (1968) on stretched exponential laws $p(k) = c\exp( -k^α)$ and apply to logarithmic hazard functions $c\exp( - (\log k)^β)$, $β>2$; they cover the big jump domain as well as the small …
Large deviation behavior of the largest eigenvalue $λ_1$ of Gaussian networks (Erdős-Rényi random graphs $\mathcal{G}_{n,p}$ with i.i.d. Gaussian weights on the edges) has been the topic of considerable interest. Recently in [6,30], a powerful approach was introduced based on tilting measures by suitable spherical integrals, particularly establishing a non-universal large deviation behavior for fixed $p<1$ compared to the standard Gaussian ($p=1$) case. The case when $p\to 0$ was however completely left open with one expecting the dense behavior to hold only until the average degree is logarit…
It has been shown by van den Berg and Steif that the sub-critical and critical Ising model on $\mathbb{Z}^d$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. …
