Mathematical physics, analysis and geometry, Mar 22, 2024
We study cover times of subsets of Z 2 by a two-dimensional massive random walk loop soup. We con... more We study cover times of subsets of Z 2 by a two-dimensional massive random walk loop soup. We consider a sequence of subsets A n ⊂ Z 2 such that |A n | → ∞ and determine the distributional limit of their cover times T (A n). We allow the killing rate κ n (or equivalently the "mass") of the loop soup to depend on the size of the set A n to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to κ −1 n = |A n | 1−8/(log log |A n |) , showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order κ −1/2 n = |A n | 1/2 , if κ −1 n exceeded |A n |, the cover times of all points in a tightly packed set A n (i.e., a square or close to a ball) would presumably be heavily correlated, complicating the analysis. Our result comes close to this extreme case.
We study spin systems defined by the winding of a random walk loop soup. For a particular choice ... more We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system sgn(ϕ) where ϕ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.
We study the critical Ising model with free boundary conditions on finite domains in Z d with d ≥... more We study the critical Ising model with free boundary conditions on finite domains in Z d with d ≥ 4. Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order |x − y| −(d−2) for large |x − y|, we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order L 2 as L → ∞. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.
Communications in Mathematical Physics, Apr 1, 2023
In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the U... more In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the Ursell functions u 2k satisfy: (−1) k−1 u 2k is increasing in each interaction. As an application, we prove a 1983 conjecture by Nishimori and Griffiths about the partition function of the Ising model with complex external field h: its closest zero to the origin (in the variable h) moves towards the origin as an arbitrary interaction increases.
We provide a representation for the scaling limit of the d = 2 critical Ising magnetization field... more We provide a representation for the scaling limit of the d = 2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d = 3 and to Potts models are also considered.
We prove Cardy's formula for rectangular crossing probabilities in dependent site percolation mod... more We prove Cardy's formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice H (with alternating updates of two sublattices) [7]; it may also be realized on the triangular lattice T with flips when a site disagrees with six, five and sometimes four of its six neighbors.
Under some general assumptions, we construct the scaling limit of open clusters and their associa... more Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling limit of the largest cluster in a bounded domain. We also apply our results to the critical, two-dimensional Ising model, obtaining the existence and uniqueness of the scaling limit of the magnetization field, as well as a geometric representation for the continuum magnetization field which can be seen as a continuum analog of the FK representation.
We introduce a natural "massive" version of the Brownian loop soup of Lawler and Werner which dis... more We introduce a natural "massive" version of the Brownian loop soup of Lawler and Werner which displays conformal covariance and exponential decay. We show that this massive Brownian loop soup arises as the near-critical scaling limit of a random walk loop soup with killing and is related to the massive SLE 2 identified by Makarov and Smirnov as the near-critical scaling limit of a loop-erased random walk with killing. We conjecture that the massive Brownian loop soup describes the zero level lines of the massive Gaussian free field, and discuss possible relations to other models, such as Ising, in the offcritical regime.
It is natural to expect that there are only three possible types of scaling limits for the collec... more It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded
We give a simple criterion for checking Edwards' hypothesis in certain zerotemperature, ferromagn... more We give a simple criterion for checking Edwards' hypothesis in certain zerotemperature, ferromagnetic spin-flip dynamics and use it to invalidate the hypothesis in various examples in dimension one and higher.
We study Mandelbrot's percolation process in dimension d ≥ 2. The process generates random fracta... more We study Mandelbrot's percolation process in dimension d ≥ 2. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0, 1] d in N d subcubes, and independently retaining or discarding each subcube with probability p or 1 − p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d − 1)-dimensional "sheets" for all d ≥ 3. For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value p c (N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0, 1] d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at p c (N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that p c (N, 2) converges, as N → ∞, to the critical density p c of site percolation on the square lattice. Assuming the existence of
ABSTRACT. We review some of the recent progress on the scaling limit of two-dimensional critical ... more ABSTRACT. We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE6 and the full scaling limit of cluster interface loops. The results given here on the full scaling ...
We investigate zero-temperature dynamics on the hexagonal lattice lHI for the homogeneous ferroma... more We investigate zero-temperature dynamics on the hexagonal lattice lHI for the homogeneous ferromagnetic Ising model with zero external magnetic field and a disordered ferromagnetic Ising model with a positive external magnetic field h. We consider both ...
We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain... more We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of $+ 1$ or $-1$ to each site in $\ mathbf {Z}^ 2$, is the zero-temperature limit ...
We analyze the geometry of scaling limits of near-critical 2D percolation, ie, for p= p c+ λδ 1/ν... more We analyze the geometry of scaling limits of near-critical 2D percolation, ie, for p= p c+ λδ 1/ν, with ν= 4/3, as the lattice spacing δ→ 0. Our proposed framework extends previous analyses for p= pc, based on SLE 6. It combines the continuum nonsimple loop ...
Abstract: We use SLE (6) paths to construct a process of continuum nonsimple loops in the plane a... more Abstract: We use SLE (6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice--that is, the scaling limit of the set of all interfaces ...
It was argued by Schramm and Smirnov that the critical site percolation exploration path on the t... more It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE 6. We provide here a detailed proof, which relies on Smirnov's theorem that crossing ...
Mathematical physics, analysis and geometry, Mar 22, 2024
We study cover times of subsets of Z 2 by a two-dimensional massive random walk loop soup. We con... more We study cover times of subsets of Z 2 by a two-dimensional massive random walk loop soup. We consider a sequence of subsets A n ⊂ Z 2 such that |A n | → ∞ and determine the distributional limit of their cover times T (A n). We allow the killing rate κ n (or equivalently the "mass") of the loop soup to depend on the size of the set A n to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to κ −1 n = |A n | 1−8/(log log |A n |) , showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order κ −1/2 n = |A n | 1/2 , if κ −1 n exceeded |A n |, the cover times of all points in a tightly packed set A n (i.e., a square or close to a ball) would presumably be heavily correlated, complicating the analysis. Our result comes close to this extreme case.
We study spin systems defined by the winding of a random walk loop soup. For a particular choice ... more We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system sgn(ϕ) where ϕ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.
We study the critical Ising model with free boundary conditions on finite domains in Z d with d ≥... more We study the critical Ising model with free boundary conditions on finite domains in Z d with d ≥ 4. Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order |x − y| −(d−2) for large |x − y|, we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order L 2 as L → ∞. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.
Communications in Mathematical Physics, Apr 1, 2023
In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the U... more In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the Ursell functions u 2k satisfy: (−1) k−1 u 2k is increasing in each interaction. As an application, we prove a 1983 conjecture by Nishimori and Griffiths about the partition function of the Ising model with complex external field h: its closest zero to the origin (in the variable h) moves towards the origin as an arbitrary interaction increases.
We provide a representation for the scaling limit of the d = 2 critical Ising magnetization field... more We provide a representation for the scaling limit of the d = 2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d = 3 and to Potts models are also considered.
We prove Cardy's formula for rectangular crossing probabilities in dependent site percolation mod... more We prove Cardy's formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice H (with alternating updates of two sublattices) [7]; it may also be realized on the triangular lattice T with flips when a site disagrees with six, five and sometimes four of its six neighbors.
Under some general assumptions, we construct the scaling limit of open clusters and their associa... more Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling limit of the largest cluster in a bounded domain. We also apply our results to the critical, two-dimensional Ising model, obtaining the existence and uniqueness of the scaling limit of the magnetization field, as well as a geometric representation for the continuum magnetization field which can be seen as a continuum analog of the FK representation.
We introduce a natural "massive" version of the Brownian loop soup of Lawler and Werner which dis... more We introduce a natural "massive" version of the Brownian loop soup of Lawler and Werner which displays conformal covariance and exponential decay. We show that this massive Brownian loop soup arises as the near-critical scaling limit of a random walk loop soup with killing and is related to the massive SLE 2 identified by Makarov and Smirnov as the near-critical scaling limit of a loop-erased random walk with killing. We conjecture that the massive Brownian loop soup describes the zero level lines of the massive Gaussian free field, and discuss possible relations to other models, such as Ising, in the offcritical regime.
It is natural to expect that there are only three possible types of scaling limits for the collec... more It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded
We give a simple criterion for checking Edwards' hypothesis in certain zerotemperature, ferromagn... more We give a simple criterion for checking Edwards' hypothesis in certain zerotemperature, ferromagnetic spin-flip dynamics and use it to invalidate the hypothesis in various examples in dimension one and higher.
We study Mandelbrot's percolation process in dimension d ≥ 2. The process generates random fracta... more We study Mandelbrot's percolation process in dimension d ≥ 2. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0, 1] d in N d subcubes, and independently retaining or discarding each subcube with probability p or 1 − p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d − 1)-dimensional "sheets" for all d ≥ 3. For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value p c (N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0, 1] d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at p c (N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that p c (N, 2) converges, as N → ∞, to the critical density p c of site percolation on the square lattice. Assuming the existence of
ABSTRACT. We review some of the recent progress on the scaling limit of two-dimensional critical ... more ABSTRACT. We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE6 and the full scaling limit of cluster interface loops. The results given here on the full scaling ...
We investigate zero-temperature dynamics on the hexagonal lattice lHI for the homogeneous ferroma... more We investigate zero-temperature dynamics on the hexagonal lattice lHI for the homogeneous ferromagnetic Ising model with zero external magnetic field and a disordered ferromagnetic Ising model with a positive external magnetic field h. We consider both ...
We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain... more We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of $+ 1$ or $-1$ to each site in $\ mathbf {Z}^ 2$, is the zero-temperature limit ...
We analyze the geometry of scaling limits of near-critical 2D percolation, ie, for p= p c+ λδ 1/ν... more We analyze the geometry of scaling limits of near-critical 2D percolation, ie, for p= p c+ λδ 1/ν, with ν= 4/3, as the lattice spacing δ→ 0. Our proposed framework extends previous analyses for p= pc, based on SLE 6. It combines the continuum nonsimple loop ...
Abstract: We use SLE (6) paths to construct a process of continuum nonsimple loops in the plane a... more Abstract: We use SLE (6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice--that is, the scaling limit of the set of all interfaces ...
It was argued by Schramm and Smirnov that the critical site percolation exploration path on the t... more It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE 6. We provide here a detailed proof, which relies on Smirnov's theorem that crossing ...
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