## Final Report Summary - LCFA (Logarithmically Correlated Fields and their Applications)

This project deals with the extremal structure of logarithmically correlated random fields. A log-correlated

field can be thought of as a random surface with certain characteristic statistics: the correlation between the

value of the field at two points grows logarithmically with the distance between the two points. Such objects

are of interest for multiple reasons. From a mathematical point of view, they are very attractive, having a rich

structure waiting to be explored. From a physical point of view, it turns out that such fields are quite typical in

nature, either in a perceptible or imperceptible form. For instance, the surface of a crystal, the interface between

two mutually repellent liquids and the energy states of a thermodynamical system are often well represented by

such fields.

Although the study of such fields, goes a long way, it is only very recently that their extremal structure

has been properly treated. By extremal structure, we mean the highest (or lowest) values of the field - the

top peaks or the lowest valleys of the surface in the crystal example, for instance. Why do we care about

understanding the statistics of such high or low values? Well, from a mathematical point of view, we strive for

a better understanding of the random behavior of such fields, for the sake of knowledge itself. From a physical

point of view, however, these extreme values play an important role in various phenomena associated with such

fields. For instance, the distribution of the state of a thermodynamical system with such an underlying potential

is governed, at certain temperatures, by the extremes of the potential.

This project builds on a fundamental result by Marek Biskup (from UCLA) and myself, where we have

characterized the macroscopic extremal structure of a particular instance in this class, known as the discrete

Gaussian free field in two dimensions (DGFF). More explicitly, we have studied how the highest values of

the DGFF behave statistically macroscopically as the size of the system grows to infinity. If one thinks of the

crystal surface example, then we have identified the probabilistic law, according to which the highest peaks

are positioned relative to each other, but only at a macroscopic scale, namely if one “looks from afar”. If ones

“looks closely” at the crystal, then one will see the local correlations. If the surface of the crystal is very high

at some point, then at nearby points it will also be very high and so high values are organized in “clusters”.

Understanding the statistics of such clusters was the onset of this project.

In thefirst few months after we started thisproject, we wereindeed successful in understanding these cluster

configurations and thereby established a full asymptotic description of the extremal structure of the DGFF.

Understanding the extremal structure of the field, opened up the door for many interesting and important new

questions. First, we wanted to understand how the underlying domain, above which the field is defined (in

the crystal example, whether the base of the crystal is a disk or a square, for instance), effects the statistics of

the extreme values. This was also accomplished shortly after. It turned out that the underlying domain enters

the description of the asymptotic behavior in the form of a random object, called the “derivative martingale”,

which possesses many interesting results and relations – of great interest by themselves. Studying the derivative

martingale was another goal in this project and so far we have derived quite a few properties of it.

Another line of research problems comes from exponentiating the field. That is, taking the exponent of the

value of the field, instead of the value itself. This has quite a few interpretations in the realm of physics and

mathematics. For instance it can model the “Gibbs-Boltzmann” distribution of the state of a thermodynamical

system whose energy landscape is represented by the field. Such, so-called “spin-glass-type systems” are very

important in physics and their study when the underlying potential is log-correlated has been so far out of reach.

In the first half of this project, we have managed to find precise asymptotics for these distributions in the case

of the DGFF, thereby confirming a conjecture about their limiting law.

Once the state distributions of such spin-glass system is sufficiently understood, the next step is to study

its evolution. Namely, how such a thermodynamical system transitions from one state to another, before or

after it has reached equilibrium. This was yet another goal in the project and we managed to make satisfactory

progress here as well. Explicitly, we can describe asymptotically the evolution of the system with respect to

the DGFF potential, both at and just before equilibrium and at low enough temperature. In particular, we show

that the system exhibits so-called “aging”, that is a gradual slow down of the rate at which it evolves.

In the second half of the project we have managed to make further and considerable (and in some cases

exhaustive) progress in the various research directions outlined above. Specifically, we have studied the geom-

etry of the clusters and its interplay with the macroscopic landscape of the extreme level-sets. This allowed

for quantitative results concerning the extreme values of the field and the resolution of various conjectures on

this topic. We have also made the identification between the derivative martingale and the critical Liouville

quantum gravity measure (LQGM), which was again expected but not proved before. Further, we have stud-

ied the sub-extremal level sets and obtained connection between their limiting law and the sub-critical LQGM

measures. Finally we have demonstrated dynamical freezing for the above mentioned spin-glass dynamics at

post-equilibrium time scales and derived a scaling limit for them, which can be seen as the natural super-critical

analog of Liouville Brownian motion.

field can be thought of as a random surface with certain characteristic statistics: the correlation between the

value of the field at two points grows logarithmically with the distance between the two points. Such objects

are of interest for multiple reasons. From a mathematical point of view, they are very attractive, having a rich

structure waiting to be explored. From a physical point of view, it turns out that such fields are quite typical in

nature, either in a perceptible or imperceptible form. For instance, the surface of a crystal, the interface between

two mutually repellent liquids and the energy states of a thermodynamical system are often well represented by

such fields.

Although the study of such fields, goes a long way, it is only very recently that their extremal structure

has been properly treated. By extremal structure, we mean the highest (or lowest) values of the field - the

top peaks or the lowest valleys of the surface in the crystal example, for instance. Why do we care about

understanding the statistics of such high or low values? Well, from a mathematical point of view, we strive for

a better understanding of the random behavior of such fields, for the sake of knowledge itself. From a physical

point of view, however, these extreme values play an important role in various phenomena associated with such

fields. For instance, the distribution of the state of a thermodynamical system with such an underlying potential

is governed, at certain temperatures, by the extremes of the potential.

This project builds on a fundamental result by Marek Biskup (from UCLA) and myself, where we have

characterized the macroscopic extremal structure of a particular instance in this class, known as the discrete

Gaussian free field in two dimensions (DGFF). More explicitly, we have studied how the highest values of

the DGFF behave statistically macroscopically as the size of the system grows to infinity. If one thinks of the

crystal surface example, then we have identified the probabilistic law, according to which the highest peaks

are positioned relative to each other, but only at a macroscopic scale, namely if one “looks from afar”. If ones

“looks closely” at the crystal, then one will see the local correlations. If the surface of the crystal is very high

at some point, then at nearby points it will also be very high and so high values are organized in “clusters”.

Understanding the statistics of such clusters was the onset of this project.

In thefirst few months after we started thisproject, we wereindeed successful in understanding these cluster

configurations and thereby established a full asymptotic description of the extremal structure of the DGFF.

Understanding the extremal structure of the field, opened up the door for many interesting and important new

questions. First, we wanted to understand how the underlying domain, above which the field is defined (in

the crystal example, whether the base of the crystal is a disk or a square, for instance), effects the statistics of

the extreme values. This was also accomplished shortly after. It turned out that the underlying domain enters

the description of the asymptotic behavior in the form of a random object, called the “derivative martingale”,

which possesses many interesting results and relations – of great interest by themselves. Studying the derivative

martingale was another goal in this project and so far we have derived quite a few properties of it.

Another line of research problems comes from exponentiating the field. That is, taking the exponent of the

value of the field, instead of the value itself. This has quite a few interpretations in the realm of physics and

mathematics. For instance it can model the “Gibbs-Boltzmann” distribution of the state of a thermodynamical

system whose energy landscape is represented by the field. Such, so-called “spin-glass-type systems” are very

important in physics and their study when the underlying potential is log-correlated has been so far out of reach.

In the first half of this project, we have managed to find precise asymptotics for these distributions in the case

of the DGFF, thereby confirming a conjecture about their limiting law.

Once the state distributions of such spin-glass system is sufficiently understood, the next step is to study

its evolution. Namely, how such a thermodynamical system transitions from one state to another, before or

after it has reached equilibrium. This was yet another goal in the project and we managed to make satisfactory

progress here as well. Explicitly, we can describe asymptotically the evolution of the system with respect to

the DGFF potential, both at and just before equilibrium and at low enough temperature. In particular, we show

that the system exhibits so-called “aging”, that is a gradual slow down of the rate at which it evolves.

In the second half of the project we have managed to make further and considerable (and in some cases

exhaustive) progress in the various research directions outlined above. Specifically, we have studied the geom-

etry of the clusters and its interplay with the macroscopic landscape of the extreme level-sets. This allowed

for quantitative results concerning the extreme values of the field and the resolution of various conjectures on

this topic. We have also made the identification between the derivative martingale and the critical Liouville

quantum gravity measure (LQGM), which was again expected but not proved before. Further, we have stud-

ied the sub-extremal level sets and obtained connection between their limiting law and the sub-critical LQGM

measures. Finally we have demonstrated dynamical freezing for the above mentioned spin-glass dynamics at

post-equilibrium time scales and derived a scaling limit for them, which can be seen as the natural super-critical

analog of Liouville Brownian motion.