Periodic Reporting for period 4 - SPRS (Stochastic Processes on Random Surfaces)
Periodo di rendicontazione: 2023-07-01 al 2024-06-30
The main focus of this research is on two-dimensional random geometry and conformally invariant probability. This area is at the interface of several different fields of mathematics, including both discrete and continuous probability theory, complex analysis, differential equations, and geometry. The main questions have to do with understanding the large-scale behavior of random discrete models which arise from statistical mechanics. One concrete example is the so-called percolation model which was first considered in the mathematics literature by Broadbent and Hammersley in 1957 and serves to model how fluid flows through a porous medium.
In the 1980s, physicists derived a number of non-rigorous predictions about the behavior of critical percolation clusters (and many other related models) in two dimensions (in which case the graph is often taken to be the planar square grid). Establishing these predictions rigorously from a mathematical perspective remained elusive, however, until work initiated by Schramm in 1999. One of Schramm's key contributions was the introduction of the Schramm-Loewner evolution (SLE), which is a mathematical tool defined by combining ideas from probability and complex analysis. It is a family of random planar curves indexed by a parameter kappa > 0 which determines its roughness; when kappa is close to 0, SLE is close to being a smooth curve and as kappa gets larger the behavior of SLE is increasingly fractal in nature. SLE for different values of kappa was subsequently shown to describe the large-scale behavior of many discrete models in two dimensions. For example, the clusters in critical percolation were proved by Smirnov to correspond to SLE with kappa=6. SLE for other values of kappa has been shown to be the scaling limit of a number of other discrete models in works of Lawler, Sheffield, Schramm, Smirnov, Werner, and others.
The methods employed in the mathematics literature, including SLE, to verify the physics predictions are in many ways quite different from the way in which the physicists originally derived their predictions. One of the key insights employed by the physicists is that models such as percolation are more tractable if one studies them on a random planar map (RPM) rather than on a deterministic graph, such as the planar square grid. Recall that a planar map is a graph together with an embedding into the plane so that the nodes are points and the edges between nodes are drawn so that they do not cross. Commonly considered cases include triangulations or quadrangulations, which respectively mean that each face has three or four adjacent edges. The properties of RPMs is itself subject of intense interest in the combinatorics literature going back to work of Tutte in the 1960s in his attempt to prove the four color theorem as well as that of Mullin on spanning-tree decorated RPMs. The subject was revitalized by bijective techniques developed in the 1980s by Cori-Vauquelin, in the 1990s by Schaeffer, and tremendous progress has been made in starting around 2000 by Angel, Chassaing, Le Gall, Marckert, Miermont, Mokkadem, Schaeffer, Schramm, and others.
When one makes a change to the underlying geometry of the graph (i.e. going from the planar square grid to a RPM), the properties of discrete models such as percolation also change. The manner in which these properties change led physicists to conjecture that the continuous object which describes the large-scale behavior of RPMs is Liouville quantum gravity (LQG). LQG is a theory of random two-dimensional Riemannian manifolds which was developed by Polyakov in the physics literature in the 1980s in the context of string theory. As a mathematical object, LQG is ill-defined, however. The interpretation of LQG as the exponential of the Gaussian free field (GFF) was developed by Duplantier-Sheffield in the 2000s and this led to the construction of its volume form; it also turns out to be related to the theory of Gaussian multiplicative chaos (GMC) developed by Kahane in the 1980s.
It has become an intense topic of research in the probability community to make the family of ideas which underlies the methods used by physicists to derive predictions for two-dimensional discrete models mathematically precise. The overall objectives for this research are centered in this direction, including establishing new scaling limit results for RPMs and connections to LQG as well as making sense of LQG in a mathematically rigorous manner (understanding it as a metric space). Altogether, the aim is to put the ideas of the physicists onto firm mathematical ground in a way which will shed further light on these random geometric structures.
In the 1980s, physicists derived a number of non-rigorous predictions about the behavior of critical percolation clusters (and many other related models) in two dimensions (in which case the graph is often taken to be the planar square grid). Establishing these predictions rigorously from a mathematical perspective remained elusive, however, until work initiated by Schramm in 1999. One of Schramm's key contributions was the introduction of the Schramm-Loewner evolution (SLE), which is a mathematical tool defined by combining ideas from probability and complex analysis. It is a family of random planar curves indexed by a parameter kappa > 0 which determines its roughness; when kappa is close to 0, SLE is close to being a smooth curve and as kappa gets larger the behavior of SLE is increasingly fractal in nature. SLE for different values of kappa was subsequently shown to describe the large-scale behavior of many discrete models in two dimensions. For example, the clusters in critical percolation were proved by Smirnov to correspond to SLE with kappa=6. SLE for other values of kappa has been shown to be the scaling limit of a number of other discrete models in works of Lawler, Sheffield, Schramm, Smirnov, Werner, and others.
The methods employed in the mathematics literature, including SLE, to verify the physics predictions are in many ways quite different from the way in which the physicists originally derived their predictions. One of the key insights employed by the physicists is that models such as percolation are more tractable if one studies them on a random planar map (RPM) rather than on a deterministic graph, such as the planar square grid. Recall that a planar map is a graph together with an embedding into the plane so that the nodes are points and the edges between nodes are drawn so that they do not cross. Commonly considered cases include triangulations or quadrangulations, which respectively mean that each face has three or four adjacent edges. The properties of RPMs is itself subject of intense interest in the combinatorics literature going back to work of Tutte in the 1960s in his attempt to prove the four color theorem as well as that of Mullin on spanning-tree decorated RPMs. The subject was revitalized by bijective techniques developed in the 1980s by Cori-Vauquelin, in the 1990s by Schaeffer, and tremendous progress has been made in starting around 2000 by Angel, Chassaing, Le Gall, Marckert, Miermont, Mokkadem, Schaeffer, Schramm, and others.
When one makes a change to the underlying geometry of the graph (i.e. going from the planar square grid to a RPM), the properties of discrete models such as percolation also change. The manner in which these properties change led physicists to conjecture that the continuous object which describes the large-scale behavior of RPMs is Liouville quantum gravity (LQG). LQG is a theory of random two-dimensional Riemannian manifolds which was developed by Polyakov in the physics literature in the 1980s in the context of string theory. As a mathematical object, LQG is ill-defined, however. The interpretation of LQG as the exponential of the Gaussian free field (GFF) was developed by Duplantier-Sheffield in the 2000s and this led to the construction of its volume form; it also turns out to be related to the theory of Gaussian multiplicative chaos (GMC) developed by Kahane in the 1980s.
It has become an intense topic of research in the probability community to make the family of ideas which underlies the methods used by physicists to derive predictions for two-dimensional discrete models mathematically precise. The overall objectives for this research are centered in this direction, including establishing new scaling limit results for RPMs and connections to LQG as well as making sense of LQG in a mathematically rigorous manner (understanding it as a metric space). Altogether, the aim is to put the ideas of the physicists onto firm mathematical ground in a way which will shed further light on these random geometric structures.
1) The Liouville quantum gravity (LQG) metric has been constructed for the range of parameters gamma in (0,2). In particular, it was shown by Ding, Dubedat, Dunlap and Falconet that certain approximations to the LQG metric admit non-trivial subsequential limits. Joint work of the PI with Gwynne subsequently showed that these subsequential limits exist as true limits which are conformally covariant and are characterized by a certain list of axioms.
2) The conformal loop ensembles (CLE) are the loop version of SLE and serve to describe the joint scaling limit of all of the interfaces in a statistical mechanics model rather than a single interface (as in the case of SLE). Joint work of the PI with Sheffield and Werner has established a number of properties of CLE on LQG which have been used to derive new formulas for critical percolation in two dimensions as well as for CLE.
3) The PI with Qian established many properties of geodesics on LQG and the Brownian map.
4) It was shown by Sheffield that the gluing of LQG surfaces along their boundary with the parameter gamma in (0,2) exists, is unique, and the gluing interface is an SLE curve (with the uniqueness building on work of Rohde-Schramm and Jones-Smirnov). Joint work of the PI with McGenteggart and Qian established a form of uniqueness in the case gamma = 2. The PI with Kavvadias and Schoug then showed that the Jones-Smirnov technique for proving conformal removability does not apply in the case gamma=2 by obtaining precise estimates for the SLE(4) uniformizing map. In the same work, the modulus of continuity of SLE(8) was obtained, proving a conjecture of Alvisio-Lawler. Finally, the PI, Kavvadias, and Schoug showed that the SLE(4) curves are conformally removable using a new technique for proving conformal removability which is adapted to random fractals. The PI, Kavvadias, and Schoug used these techniques to prove the removability of SLE(kappa) for kappa in (4,8) when the graph of components is connected.
5) The PI started the program of constructing a metric for CLE(kappa) by proving tightness for an approximation scheme when (8/3,4). The PI together with Ambrosio and Yuan proved the analogous result for kappa in (4,8).
6) The PI together with Andres, Kajino, and Kavvadias have established on and off-diagonal heat kernel estimates for Liouville Brownian motion when gamma = sqrt(8/3).
2) The conformal loop ensembles (CLE) are the loop version of SLE and serve to describe the joint scaling limit of all of the interfaces in a statistical mechanics model rather than a single interface (as in the case of SLE). Joint work of the PI with Sheffield and Werner has established a number of properties of CLE on LQG which have been used to derive new formulas for critical percolation in two dimensions as well as for CLE.
3) The PI with Qian established many properties of geodesics on LQG and the Brownian map.
4) It was shown by Sheffield that the gluing of LQG surfaces along their boundary with the parameter gamma in (0,2) exists, is unique, and the gluing interface is an SLE curve (with the uniqueness building on work of Rohde-Schramm and Jones-Smirnov). Joint work of the PI with McGenteggart and Qian established a form of uniqueness in the case gamma = 2. The PI with Kavvadias and Schoug then showed that the Jones-Smirnov technique for proving conformal removability does not apply in the case gamma=2 by obtaining precise estimates for the SLE(4) uniformizing map. In the same work, the modulus of continuity of SLE(8) was obtained, proving a conjecture of Alvisio-Lawler. Finally, the PI, Kavvadias, and Schoug showed that the SLE(4) curves are conformally removable using a new technique for proving conformal removability which is adapted to random fractals. The PI, Kavvadias, and Schoug used these techniques to prove the removability of SLE(kappa) for kappa in (4,8) when the graph of components is connected.
5) The PI started the program of constructing a metric for CLE(kappa) by proving tightness for an approximation scheme when (8/3,4). The PI together with Ambrosio and Yuan proved the analogous result for kappa in (4,8).
6) The PI together with Andres, Kajino, and Kavvadias have established on and off-diagonal heat kernel estimates for Liouville Brownian motion when gamma = sqrt(8/3).
The work mentioned above all represents progress beyond the state of the art, the main piece of progress being the construction of the LQG metric for gamma in (0,2).