# LARGEDEVRWRE — Result In Brief

Project ID:
322078

Funded under:
FP7-PEOPLE

Country:
Turkey

Domain:
Fundamental Research

## A walk on the wild side

An EU-funded initiative carefully studied one of the standard models used in the study of random media – random walk in a random environment (RWRE).

© Silent Corners, Shutterstock

Large deviation theory is a cornerstone of modern probability and used in statistical mechanics, information theory, risk management plus many other fields. The project LARGEDEVRWRE (Large deviations for random walks in random environments) was established to achieve a deeper understanding of the large deviation properties of multidimensional RWRE.

Many natural systems demonstrate complex phenomena, which can be studied using stochastic models that often include a ‘tuneable’ parameter such as time, volume, or number of particles. Some of these models are shown to behave, either deterministically or in a random but well understood way, as this parameter is taken to its extreme value (either zero or infinity).

Such results are referred to as limit theorems, which include laws of large numbers (LLNs). An LLN is often accompanied by a central limit theorem and/or large deviation principle. The term ‘large deviation’ is derived from the fact that a complementary rare event is a large deviation from typical behaviour.

Researchers studied the large deviation properties of a series of models to obtain simple expressions for the quenched and averaged large deviation rate functions. The rate functions were compared to discover a formula that links them. Large deviation results and techniques gave precise results for variational formulas that were simpler than those existing in the literature for characterising the disorder level of the random environment.

The project made a significant impact on probability theory through its unified approach to random media models and the development of robust techniques. This unified approach was based on taking the point of view of the particle performing the random motion (first layer) in the random environment (second layer), and reducing the two layers of randomness to one.

LARGEDEVRWRE studied other models with two layers of randomness, namely stochastic encounter-mating models from population dynamics. These models will help to increase the scientific understanding of evolution, the dynamics of sexually transmitted diseases and can also be applied to computer science and economics.

Many natural systems demonstrate complex phenomena, which can be studied using stochastic models that often include a ‘tuneable’ parameter such as time, volume, or number of particles. Some of these models are shown to behave, either deterministically or in a random but well understood way, as this parameter is taken to its extreme value (either zero or infinity).

Such results are referred to as limit theorems, which include laws of large numbers (LLNs). An LLN is often accompanied by a central limit theorem and/or large deviation principle. The term ‘large deviation’ is derived from the fact that a complementary rare event is a large deviation from typical behaviour.

Researchers studied the large deviation properties of a series of models to obtain simple expressions for the quenched and averaged large deviation rate functions. The rate functions were compared to discover a formula that links them. Large deviation results and techniques gave precise results for variational formulas that were simpler than those existing in the literature for characterising the disorder level of the random environment.

The project made a significant impact on probability theory through its unified approach to random media models and the development of robust techniques. This unified approach was based on taking the point of view of the particle performing the random motion (first layer) in the random environment (second layer), and reducing the two layers of randomness to one.

LARGEDEVRWRE studied other models with two layers of randomness, namely stochastic encounter-mating models from population dynamics. These models will help to increase the scientific understanding of evolution, the dynamics of sexually transmitted diseases and can also be applied to computer science and economics.