The Variable Range Hopping is considered in the Physics literature as an effective model for the analysis of conductivity in semiconductors. Understanding how the macroscopic parameters depend on the small-scale randomness of the environment and proving the Einstein Relation for this model is the ambitious aim of this project. Main objectives: 1) Extend recent results (law of large numbers, existence of a stationary state) for long-range reversible random walks on point processes including the possibility of traps. 2) Analyze how an external field influences the limiting velocity of the Variable Range Hop- ping, in comparison to similar models from Mathematical Physics. 3) Establish the first rigorous Einstein Relation for a physically relevant model, the Variable Range Hopping. The mathematical techniques we have at our disposal nowadays (such as the weak Einstein Relation and the control of long range models) are a solid basis for the investigation of the problem: This would be the first time an Einstein Relation is rigorously proven for a relevant physical model. Furthermore, the richness of the subject guarantees also many intermediate results of great relevance in the field of Probability Theory. Besides the big scientific relevance of the expected results, the project will have a strong impact also on the career of the experienced researcher, completing his international profile of independent scientist, and will also strengthen the interplay between the Probability Theory communities of France, Germany and Italy. Finally, a positive outcome of the action will bring a significant insight on the physical study of semiconductors.

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to the maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold known as the mixing time. Discovered four decades ago in the context of card shuffling, this dynamical phase transition has since then been observed in a variety of situations, from random walks on random graphs to high-temperature spin glasses. It is now believed to be universal among fast-mixing high-dimensional systems. Yet, the current proofs are case-specific and rely on explicit computations which (i) can only be carried out in oversimplified models and (ii) do not bring any conceptual insight as to why such a sharp transition occurs. Our ambition here is to identify the general conditions that trigger the cutoff phenomenon. This is one of the biggest challenges in the quantitative analysis of finite Markov chains. We believe that the key is to harness a new information-theoretic statistics called varentropy, whose relevance was recently uncovered by the PI but whose systematic estimation remains entirely to be developed. Specifically, we propose to elaborate a robust set of analytic and geometric tools to bound varentropy and control its evolution under any Markov semi-group, much like log-Sobolev inequalities do for entropy. From this, we intend to derive sharp and easily verifiable criteria allowing us to predict cutoff without having to compute mixing times. If successful, our approach will not only provide a unified explanation for all known instances of the phenomenon, but also confirm its long-predicted occurrence in a number of models of fundamental importance. Emblematic applications include random walks on expanders, interacting particle systems, and MCMC algorithms.

Equity in education is a central and long-standing concern of countries around the world. Despite significant amounts of public resources invested over several decades towards disadvantaged children, the relationship between education outcomes and personal circumstances remains persistently strong in most countries. The objectives of my project EMANCIPATE is to make progress in our understanding of why there is such persistent social inequity in education. I will explore a new channel: perceptions and beliefs resulting from social stereotypes and leading to low confidence, lack of hope, and discouragement. I argue that how students perceive their capacity and the likelihood that they will succeed is an important driver of their motivation and effort, which might be just as important for academic performance as external factors such as class size or teacher salaries. Self-perceptions and anticipations may thus be one way social disadvantage perpetuates itself. The key questions that I will address are whether socioeconomic status creates low confidence, lack of hope, and resignation, whether this affects downstream educational attainment, labour market outcomes, and individual welfare, and how this negative feedback loop can be broken. EMANCIPATE will advance this research in three steps. First, I will use international assessment tools on nationally representative samples of adolescents to test the accuracy of their self-perceptions and anticipations, and the existence and scope of the resignation mindset. Second, I will use two interventions aimed at reducing the resignation mindset to examine its long-term impacts on educational attainment, labour market outcomes, and individual welfare. Third, I will develop and test two interventions targeting the main actors of the education system: the teachers and the parents, to foster students’ self-confidence, hope, and motivation, and ultimately reduce the achievement gap between disadvantaged and advantaged students.