## Final Report Summary - LARGEDEVRWRE (Large Deviations for Random Walks in Random Environments)

Many natural systems exhibit complex phenomena and are analyzed using stochastic models which often involve a tunable parameter, e.g., time, volume, number of particles, noise level. Some of these models, when viewed at appropriate scales, are shown to behave (asymptotically) either deterministically or in a random but well-understood way, as this parameter is taken to its extreme value (typically zero or infinity). Such results are generally referred to as limit theorems. Fundamental examples of limit theorems include laws of large numbers (LLNs) and ergodic theorems which are stated at the scale at which the model behaves deterministically, with probability approaching one as the parameter is taken to its extreme value.

An LLN is often accompanied by a central limit theorem and/or a large deviation principle. The former considers a finer scale at which the random fluctuations around the deterministic behavior are described by a Gaussian distribution, whereas the latter identifies the exponential rate with which the probability of the event in the statement of the LLN approaches one. The term “large deviation” comes from the fact that the complementary rare event (whose probability converges exponentially to zero) is a large deviation from the typical behavior.

In this project, the researcher Yilmaz and his collaborators studied various models of random media that are closely related to each other:

(i) random walk in random environment,

(ii) random walk in dynamic random environment,

(iii) diffusion in random environment,

(iv) directed polymer in random environment,

(v) random walk in random potential,

(vi) discrete Bellman equation with random non-convex Hamiltonian,

(vii) Hamilton-Jacobi-Bellman equation with random convex Hamiltonian.

In collaboration with Rassoul-Agha (University of Utah) and Seppalainen (UW Madison), Yilmaz closely analyzed the large deviation properties of the models (i-iii). They answered questions regarding how the exponential rate of decay of the probabilities or rare events regarding the random motion depends on averaging vs. quenching the random environment, how these models behave when they are conditioned on these rare events, and what the environment empirically looks like from the point of view of the particle performing the motion under such a conditioning. The answers involve characterizations via variational formulas, a wide array of entropy functions, convex optimization, and ergodic changes of measure via generalized harmonic functions.

The aforementioned large deviation results and the techniques used in their proofs give certain laws of large numbers for the remaining models. Indeed, Yilmaz and his previously mentioned collaborators carried out this program and obtained precise results about the typical behavior of models (iv,v). In particular, they gave variational formulas for the so-called quenched free energy of these models (i.e., the net result of the competition between energy and entropy), which are simpler than the existing ones in the literature. They used these formulas to characterize the disorder level of the random environment. They also exactly solved these variational problems when the random environment is constructed in a specific way by independent gamma distributed random variables.

Similarly, in collaboration with Zeitouni (Weizmann Institute) and Rezakhanlou (UC Berkeley), Yilmaz solved certain technical problems regarding the homogenization of equations of the type (vi,vii), which means that they behave like equations with deterministic Hamiltonians when time and space variables are scaled appropriately. Homogenization can be viewed as an LLN, yet for the specific family of equations Yilmaz et al. worked on, it can be approached via a large deviation analysis of certain underlying processes which are of the type (i-iii).

Models of random media are important not only because they are frequently used in statistical mechanics, fluid dynamics and materials science, but also because their mathematical analysis is challenging. Indeed, random media problems are at the frontier of probability theory as they require novel methods. The impact of the project lies in its unified approach to these models, the development of robust techniques that are applicable to many of them, and obtaining results that have the potential to shed light on long-standing open problems. This unified approach is built on taking the point of view of the particle performing the random motion (first layer) in the random environment (second layer), and thereby reducing the two layers of randomness to one, albeit at the cost of working in greatly enlarged spaces. In other words, this approach brings the tools of infinite dimensional analysis into discrete probability.

The project eventually branched out to studying another family of models with two layers of randomness, namely stochastic encounter-mating models from population dynamics. These models describe the monogamous permanent pair formation of animals with multitype females and males. The two layers consist of random encounters (governed by random firing times of the animals) and random matings (governed by their mating preferences). Yilmaz and his collaborator Gun (WIAS Berlin) put these models in a unified mathematical framework and characterized (via the so-called fine balance condition) when their resulting mating patterns exhibit zero correlation (i.e., panmixia) between female and male types, settling a long-standing conjecture of Gimelfarb (1988). In the case with two types, they gave a complete characterization of positive/zero/negative correlation (i.e., assortative mating).

Another aspect of Yilmaz’s work on stochastic encounter-mating that is parallel to his work on random media is the establishment of limit theorems. Indeed, Gun and Yilmaz proved various LLNs and central limit theorems for these models as the population size diverges to infinity. In particular, in the infinite population, the density of pairs of given types evolve according to a system of coupled ordinary differential equations. Gun and Yilmaz solved these equations under the fine balance condition as well as under certain symmetry conditions when there are only two types.

In the biology literature, panmixia is often referred to as random mating. However, sometimes it is not made clear if one merely means panmixia or no mating preferences. Gun and Yilmaz’s work rigorously proved a claim by Gimelfarb (1988) that there are nontrivial mating preferences resulting in panmixia. This has significant impact in experimental biology, in a negative way: Mating preferences cannot be inferred from mating patterns. Moreover, if properly generalized to include separation and/or offspring production, stochastic encounter-mating models have the potential to improve our understanding of sexual evolution and the dynamics of sexually transmitted diseases. Finally, they can be used to analyze other two-sided matching problems, e.g., in the context of computer science and economics.

The results of this project have been published (or soon will be published) as research articles which are accessible from Yilmaz’s website: http://home.ku.edu.tr/~atillayilmaz/

An LLN is often accompanied by a central limit theorem and/or a large deviation principle. The former considers a finer scale at which the random fluctuations around the deterministic behavior are described by a Gaussian distribution, whereas the latter identifies the exponential rate with which the probability of the event in the statement of the LLN approaches one. The term “large deviation” comes from the fact that the complementary rare event (whose probability converges exponentially to zero) is a large deviation from the typical behavior.

In this project, the researcher Yilmaz and his collaborators studied various models of random media that are closely related to each other:

(i) random walk in random environment,

(ii) random walk in dynamic random environment,

(iii) diffusion in random environment,

(iv) directed polymer in random environment,

(v) random walk in random potential,

(vi) discrete Bellman equation with random non-convex Hamiltonian,

(vii) Hamilton-Jacobi-Bellman equation with random convex Hamiltonian.

In collaboration with Rassoul-Agha (University of Utah) and Seppalainen (UW Madison), Yilmaz closely analyzed the large deviation properties of the models (i-iii). They answered questions regarding how the exponential rate of decay of the probabilities or rare events regarding the random motion depends on averaging vs. quenching the random environment, how these models behave when they are conditioned on these rare events, and what the environment empirically looks like from the point of view of the particle performing the motion under such a conditioning. The answers involve characterizations via variational formulas, a wide array of entropy functions, convex optimization, and ergodic changes of measure via generalized harmonic functions.

The aforementioned large deviation results and the techniques used in their proofs give certain laws of large numbers for the remaining models. Indeed, Yilmaz and his previously mentioned collaborators carried out this program and obtained precise results about the typical behavior of models (iv,v). In particular, they gave variational formulas for the so-called quenched free energy of these models (i.e., the net result of the competition between energy and entropy), which are simpler than the existing ones in the literature. They used these formulas to characterize the disorder level of the random environment. They also exactly solved these variational problems when the random environment is constructed in a specific way by independent gamma distributed random variables.

Similarly, in collaboration with Zeitouni (Weizmann Institute) and Rezakhanlou (UC Berkeley), Yilmaz solved certain technical problems regarding the homogenization of equations of the type (vi,vii), which means that they behave like equations with deterministic Hamiltonians when time and space variables are scaled appropriately. Homogenization can be viewed as an LLN, yet for the specific family of equations Yilmaz et al. worked on, it can be approached via a large deviation analysis of certain underlying processes which are of the type (i-iii).

Models of random media are important not only because they are frequently used in statistical mechanics, fluid dynamics and materials science, but also because their mathematical analysis is challenging. Indeed, random media problems are at the frontier of probability theory as they require novel methods. The impact of the project lies in its unified approach to these models, the development of robust techniques that are applicable to many of them, and obtaining results that have the potential to shed light on long-standing open problems. This unified approach is built on taking the point of view of the particle performing the random motion (first layer) in the random environment (second layer), and thereby reducing the two layers of randomness to one, albeit at the cost of working in greatly enlarged spaces. In other words, this approach brings the tools of infinite dimensional analysis into discrete probability.

The project eventually branched out to studying another family of models with two layers of randomness, namely stochastic encounter-mating models from population dynamics. These models describe the monogamous permanent pair formation of animals with multitype females and males. The two layers consist of random encounters (governed by random firing times of the animals) and random matings (governed by their mating preferences). Yilmaz and his collaborator Gun (WIAS Berlin) put these models in a unified mathematical framework and characterized (via the so-called fine balance condition) when their resulting mating patterns exhibit zero correlation (i.e., panmixia) between female and male types, settling a long-standing conjecture of Gimelfarb (1988). In the case with two types, they gave a complete characterization of positive/zero/negative correlation (i.e., assortative mating).

Another aspect of Yilmaz’s work on stochastic encounter-mating that is parallel to his work on random media is the establishment of limit theorems. Indeed, Gun and Yilmaz proved various LLNs and central limit theorems for these models as the population size diverges to infinity. In particular, in the infinite population, the density of pairs of given types evolve according to a system of coupled ordinary differential equations. Gun and Yilmaz solved these equations under the fine balance condition as well as under certain symmetry conditions when there are only two types.

In the biology literature, panmixia is often referred to as random mating. However, sometimes it is not made clear if one merely means panmixia or no mating preferences. Gun and Yilmaz’s work rigorously proved a claim by Gimelfarb (1988) that there are nontrivial mating preferences resulting in panmixia. This has significant impact in experimental biology, in a negative way: Mating preferences cannot be inferred from mating patterns. Moreover, if properly generalized to include separation and/or offspring production, stochastic encounter-mating models have the potential to improve our understanding of sexual evolution and the dynamics of sexually transmitted diseases. Finally, they can be used to analyze other two-sided matching problems, e.g., in the context of computer science and economics.

The results of this project have been published (or soon will be published) as research articles which are accessible from Yilmaz’s website: http://home.ku.edu.tr/~atillayilmaz/