We consider a random walk in a uniformly elliptic i.i.d. random environment in Z d for d ≥ 2. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, in [Sz01, Sz02], Sznitman defined the so called conditions (T) and (T ). The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width L. The second one is the requirement that for all γ ∈ (0, 1) condition (T) γ is satisfied, which in turn is defined as the requirement that the decay is like e −CL γ for some C > 0. In this article we prove a conjecture of Sznitman of 2002 [Sz02], stating that (T) and (T ) are equivalent. Hence, this closes the circle proving the equivalence of conditions (T), (T ) and (T) γ for some γ ∈ (0, 1) as conjectured in [Sz02], and also of each of these ballisticity conditions with the polynomial condition (P) M for M ≥ 15d + 5 introduced by Berger, Drewitz and Ramírez in [BDR14].

We prove that every random walk in a uniformly elliptic random environment satisfying the cone mixing condition and a non-effective polynomial ballisticity condition with high enough degree has an asymptotic direction.

It is conjectured that in dimensions d ≥ 2 any random walk in an i.i.d. uniformly elliptic random environment (RWRE) which is directionally transient is ballistic. The ballisticity conditions for RWRE somehow interpolate between directional transience and ballisticity and have served to quantify the gap which would need to be proven in order to answer affirmatively this conjecture. Two important ballisticity conditions introduced by Sznitman [Sz02] in 2001 and 2002 are the so called conditions (T ′ ) and (T ): given a slab of width L orthogonal to l, condition (T ′ ) in direction l is the requirement that the annealed exit probability of the walk through the side of the slab in the half-space {x : x·l < 0}, decays faster than e −CL γ for all γ ∈ (0, 1) and some constant C > 0, while condition (T ) in direction l is the requirement that the decay is exponential e −CL . It is believed that (T ′ ) implies (T ). In this article we show that (T ′ ) implies at least an almost (in a sense to be made precise) exponential decay.

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Material property viscoelastic inversion studies often rely on the continuous -time framework for Fourier analysis, which may not accurately represent real experimentally collected data. In this paper, we address the discrete and finite nature of viscoelastic functions obtained from experiments and discuss the impact of these characteristics on the frequency spectrum analysis. We derive equations for the Discrete-Time Fourier Transform (DTFT) of a discrete-finite stress relaxation signal corresponding to the relaxation of a generalized Maxwell model. Our analysis highlights the limitations of the traditional continuous -time framework in capturing the inherent features of real signals, which are discrete and finite in nature. This results in two phenomena: aliasing and spectral leakage. We present equations that consider these phenomena, allowing experimentalists to anticipate and account for aliasing and leakage when performing model fitting. The proposed discrete-finite approach provides a more accurate representation of real viscoelastic data, enabling researchers to make better-informed decisions in the analysis and interpretation of sample viscoelastic functions.