Thomas G Kennedy

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• Kennedy, T. G., & Lawler, G. F. (2013). Lattice effects in the scaling limit of the two-dimensional self-avoiding walk.. In Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics (2011)(pp 195-210). American Mathematical Society.

Journals/Publications

• Kennedy, T. G. (2022). Tensor RG approach to high-temperature fixed point. J. Stat. Phys., 187, 33.
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  We study a renormalization group (RG) map for tensor networks that include two-dimensional lattice spin systems such as the Ising model. Numerical studies of such RG maps have been quite successful at reproducing the known critical behavior. In those numerical studies the RG map must be truncated to keep the dimension of the legs of the tensors bounded. Our tensors act on an infinite-dimensional Hilbert space, and our RG map does not involve any truncations. Our RG map has a trivial fixed point which represents the high-temperature fixed point. We prove that if we start with a tensor that is close to this fixed point tensor, then the iterates of the RG map converge in the Hilbert-Schmidt norm to the fixed point tensor. It is important to emphasize that this statement is not true for the simplest tensor network RG map in which one simply contracts four copies of the tensor to define the renormalized tensor. The linearization of this simple RG map about the fixed point is not a contraction due to the presence of so-called CDL tensors. Our work provides a first step towards the important problem of the rigorous study of RG maps for tensor networks in a neighborhood of the critical point. [Journal_ref: ]
• Kennedy, T., & Rychkov, S. (2024). Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point. Ann. Henri Poincare.
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  We continue our study of rigorous renormalization group (RG) maps for tensor networks that was begun in arXiv:2107.11464. In this paper we construct a rigorous RG map for 2D tensor networks whose domain includes tensors that represent the 2D Ising model at low temperatures with a magnetic field $h$. We prove that the RG map has two stable fixed points, corresponding to the two ground states, and one unstable fixed point which is an example of a discontinuity fixed point. For the Ising model at low temperatures the RG map flows to one of the stable fixed points if $h \neq 0$, and to the discontinuity fixed point if $h=0$. In addition to the nearest neighbor and magnetic field terms in the Hamiltonian, we can include small terms that need not be spin-flip invariant. In this case we prove there is a critical value $h_c$ of the field (which depends on these additional small interactions and the temperature) such that the RG map flows to the discontinuity fixed point if $h=h_c$ and to one of the stable fixed points otherwise. We use our RG map to give a new proof of previous results on the first-order transition, namely, that the free energy is analytic for $h \neq h_c$, and the magnetization is discontinuous at $h = h_c$. The construction of our low temperature RG map, in particular the disentangler, is surprisingly very similar to the construction of the map in arXiv:2107.11464 for the high temperature phase. We also give a pedagogical discussion of some general rigorous transformations for infinite dimensional tensor networks and an overview of the proof of stability of the high temperature fixed point for the RG map in arXiv:2107.11464. [Journal_ref: ]
• Kennedy, T. G. (2020). Absence of renormalization group pathologies in some critical Dyson-Ising ferromagnets. Preprint.
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  The Dyson-Ising ferromagnet is a one-dimensional Ising model with a power lawinteraction. When the power is between -1 and -2, the model has a phasetransition. Van Enter and Le Ny proved that at sufficiently low temperaturesthe decimation renormalization group transformation is not defined in the sensethat the renormalized measure is not a Gibbs measure. We consider a modifiedmodel in which the nearest neighbor couplings are much larger than the othercouplings. For a family of Hamiltonians which includes critical cases, we provethat the first step of the renormalization group transformation can berigorously defined for majority rule and decimation.[Journal_ref: ]
• Kennedy, T. (2019). A Non-intersecting Random Walk on the Manhattan Lattice and {SLE} 6. Journal of Statistical Physics, 174(1), 77-96. doi:10.1007/s10955-018-2176-9
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  We consider a random walk on the Manhattan lattice. The walker must follow the orientations of the bonds in this lattice, and the walker is not allowed to visit a site more than once. When both possible steps are allowed, the walker chooses between them with equal probability. The walks generated by this model are known to be related to interfaces for bond percolation on a square lattice. So it is natural to conjecture that the scaling limit is $$\hbox {SLE}_6$$ . We test this conjecture with Monte Carlo simulations of the random walk model and find strong support for the conjecture.
• Kennedy, T. (2019). Conformal invariance of the loop-erased percolation explorer. J. Stat. Phys., 177, 1-19. doi:10.1007/s10955-019-02354-9
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  We consider critical percolation on the triangular lattice in a boundedsimply connected domain with boundary conditions that force an interfacebetween two prescribed boundary points. We say the interface forms a"near-loop" when it comes within one lattice spacing of itself. We define a newcurve by erasing these near-loops as we traverse the interface. Our Monte Carlosimulations of this model lead us to conclude that the scaling limit of thisloop-erased percolation interface is conformally invariant and has fractaldimension 4/3. However, it is not SLE_8/3. We also consider the process inwhich a near-loop is when the explorer comes within two lattice spacings ofitself.[Journal_ref: ]
• Kennedy, T. G. (2018). A non-intersecting random walk on the Manhattan lattice and SLE_6. J. Statist. Phys., Appeared on-line, not in print yet. doi:https://doi.org/10.1007/s10955-018-2176-9
• Kennedy, T. G. (2017). The difference between a discrete and continuous harmonic measure. J. Theoret. Probab..
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  We consider a discrete-time, continuous-state random walk with stepsuniformly distributed in a disk of radius of $h$. For a simply connected domain$D$ in the plane, let $\omega_h(0,\cdot;D)$ be the discrete harmonic measure at$0\in D$ associated with this random walk, and $\omega(0,\cdot;D)$ be the(continuous) harmonic measure at $0$. For domains $D$ with analytic boundary,we prove there is a bounded continuous function $\sigma_D(z)$ on $\partial D$such that for functions $g$ which are in $C^{2+\alpha}(\partial D)$ for some$\alpha>0$ $$ \lim_{h\downarrow 0} \frac{\int_{\partial D} g(\xi)\omega_h(0,|d\xi|;D) -\int_{\partial D} g(\xi)\omega(0,|d\xi|;D)}{h} =\int_{\partial D}g(z) \sigma_D(z) |dz|. $$ We give an explicit formula for$\sigma_D$ in terms of the conformal map from $D$ to the unit disc. The proofrelies on some fine approximations of the potential kernel and Green's functionof the random walk by their continuous counterparts, which may be ofindependent interest.[Journal_ref: ]
• Kennedy, T. G. (2016). The First Order Correction to the Exit Distribution for Some Random Walks. JOURNAL OF STATISTICAL PHYSICS, 164(1), 174-189.
• Kennedy, T. (2015). Conformal Invariance Predictions for the Three-Dimensional Self-Avoiding Walk. JOURNAL OF STATISTICAL PHYSICS, 158(6), 1195-1212.
• Kennedy, T. (2015). The Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution. JOURNAL OF STATISTICAL PHYSICS, 160(2), 302-320.
• Guttmann, A. J., & Kennedy, T. (2014). Self-avoiding walks in a rectangle. Journal of Engineering Mathematics, 84(1), 201-208. doi:10.1007/s10665-013-9622-0
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  A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a \(10\,\times \,1\) rectangle and to evaluate the probability of a Brownian path hitting the short ends of the rectangle before hitting one of the long sides. For Brownian motion this probability can be calculated exactly (The SIAM 100-digit challenge: a study in high-accuracy numerical computing. SIAM, Philadelphia, 2004). Here we consider instead the more difficult problem of a self-avoiding walk (SAW) in the scaling limit and pose the same question. Assuming that the scaling limit of SAW is conformally invariant, we evaluate, asymptotically, the same probability. For the SAW case we find the probability is approximately 200 times greater than for Brownian motion.
• Guttmann, A. J., & Kennedy, T. (2013). Self-avoiding walks in a rectangle. Journal of Engineering Mathematics, 1-8.
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  Abstract: A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a {Mathematical expression} rectangle and to evaluate the probability of a Brownian path hitting the short ends of the rectangle before hitting one of the long sides. For Brownian motion this probability can be calculated exactly (The SIAM 100-digit challenge: a study in high-accuracy numerical computing. SIAM, Philadelphia, 2004). Here we consider instead the more difficult problem of a self-avoiding walk (SAW) in the scaling limit and pose the same question. Assuming that the scaling limit of SAW is conformally invariant, we evaluate, asymptotically, the same probability. For the SAW case we find the probability is approximately 200 times greater than for Brownian motion. © 2013 Springer Science+Business Media Dordrecht.
• Kennedy, T. (2013). Conformal invariance of the 3D self-avoiding walk. Physical Review Letters, 111(16).
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  Abstract: We show that if the three-dimensional self-avoiding walk (SAW) is conformally invariant, then one can compute the hitting densities for the SAW in a half-space and in a sphere. We test these predictions by Monte Carlo simulations and find excellent agreement, thus providing evidence that the SAW is conformally invariant in three dimensions. © 2013 American Physical Society.
• Kennedy, T. (2012). Efficient SLE algorithms and numerical pitfalls of the method. Bulletin of the American Physical Society, 2012.
• Kennedy, T. (2012). Simulating self-avoiding walks in bounded domains. Journal of Mathematical Physics, 53(9).
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  Abstract: Let D be a domain in the plane containing the origin. We are interested in the ensemble of self-avoiding walks (SAWs) in D which start at the origin and end on the boundary of the domain. We introduce an ensemble of SAWs that we expect to have the same scaling limit. The advantage of our ensemble is that it can be simulated using the pivot algorithm. Our ensemble makes it possible to accurately study Schramm-Loewner evolution (SLE) predictions for the SAW in bounded simply connected domains. One such prediction is the distribution along the boundary of the endpoint of the SAW. We use the pivot algorithm to simulate our ensemble and study this density. In particular the lattice effects in this density that persist in the scaling limit are seen to be given by a purely local function. © 2012 American Institute of Physics.
• Kennedy, T. (2012). Transforming Fixed-Length Self-avoiding Walks into Radial SLE_8/3. Journal of Statistical Physics, 146(2), 281-293.
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  Abstract: We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE8/3. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE8/3, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values. © 2011 Springer Science+Business Media, LLC.
• Dyhr, B., Gilbert, M., Kennedy, T., Lawler, G. F., & Passon, S. (2011). The Self-avoiding Walk Spanning a Strip. Journal of Statistical Physics, 144(1), 1-22.
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  Abstract: We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as β → βc- of the probability measure on all finite length walks ω with the probability of ω proportional to β{pipe}ω{pipe} where {pipe}ω{pipe} is the number of steps in ω. (βc is the reciprocal of the connective constant.) The self-avoiding walk in a strip {z : 0 < Im(z) < y} is defined by considering all self-avoiding walks ω in the strip which start at the origin and end somewhere on the top boundary with probability proportional to βc{pipe}ω{pipe}. We prove that this probability measure may be obtained by conditioning the SAW in the half plane to have a bridge at height y. This observation is the basis for simulations to test conjectures on the distribution of the endpoint of the SAW in a strip and the relationship between the distribution of this strip SAW and SLE8/3. © 2011 Springer Science+Business Media, LLC.
• Kennedy, T. (2010). Renormalization group maps for ising models in lattice-gas variables. Journal of Statistical Physics, 140(3), 409-426.
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  Abstract: Real-space renormalization group maps, e.g., the majority rule transformation, map Ising-type models to Ising-type models on a coarser lattice. We show that each coefficient in the renormalized Hamiltonian in the lattice-gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice-gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice-gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them. © 2010 Springer Science+Business Media, LLC.
• Bauer, M., Bernard, D., & Kennedy, T. (2009). Conditioning schramm-loewner evolutions and loop erased random walks. Journal of Mathematical Physics, 50(4).
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  Abstract: We discuss properties of dipolar Schramm-Loewner evolution (SLEκ) under conditioning. We show that κ=2, which describes continuum limits of loop erased random walks, is characterized as being the only value of κ such that dipolar SLE conditioned to stop on an interval coincides with dipolar SLE on that interval. We illustrate this property by computing a new bulk passage probability for SLE2. © 2009 American Institute of Physics.
• Kennedy, T. (2009). Numerical computations for the schramm-loewner evolution. Journal of Statistical Physics, 137(5), 839-856.
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  Abstract: We review two numerical methods related to the Schramm-Loewner evolution (SLE). The first simulates SLE itself. More generally, it finds the curve in the half-plane that results from the Loewner equation for a given driving function. The second method can be thought of as the inverse problem. Given a simple curve in the half-plane it computes the driving function in the Loewner equation. This algorithm can be used to test if a given random family of curves in the half-plane is SLE by computing the driving process for the curves and testing if it is Brownian motion. More generally, this algorithm can be used to compute the driving process for random curves that may not be SLE. Most of the material presented here has appeared before. Our goal is to give a pedagogic review, illustrate some of the practical issues that arise in these computations and discuss some open problems. © Springer Science+Business Media, LLC 2009.
• Kennedy, T. (2008). Computing the loewner driving process of random curves in the half plane. Journal of Statistical Physics, 131(5), 803-819.
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  Abstract: We simulate several models of random curves in the half plane and numerically compute the stochastic driving processes that produce the curves through the Loewner equation. Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion, as it is for SLE. We find that testing only the normality of the process at a fixed time is not effective at determining if the random curves are an SLE. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N 1.35) rather than the usual O(N 2), where N is the number of points on the curve. © 2008 Springer Science+Business Media, LLC.
• Kennedy, T. (2007). A fast algorithm for simulating the chordal Schramm-Loewner Evolution. Journal of Statistical Physics, 128(5), 1125-1137.
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  Abstract: The Schramm-Loewner evolution (SLE) can be simulated by dividing the time interval into N subintervals and approximating the random conformal map of the SLE by the composition of N random, but relatively simple, conformal maps. In the usual implementation the time required to compute a single point on the SLE curve is O(N). We give an algorithm for which the time to compute a single point is O(N p ) with p
• Kennedy, T. (2007). The length of an SLE-Monte Carlo studies. Journal of Statistical Physics, 128(6), 1263-1277.
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  Abstract: The scaling limits of a variety of critical two-dimensional lattice models are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the parameter κ. These lattice models have a natural parametrization of their random curves given by the length of the curve. This parametrization (with suitable scaling) should provide a natural parametrization for the curves in the scaling limit. We conjecture that this parametrization is also given by a type of fractal variation along the curve, and present Monte Carlo simulations to support this conjecture. Then we show by simulations that if this fractal variation is used to parametrize the SLE, then the parametrized curves have the same distribution as the curves in the scaling limit of the lattice models with their natural parametrization. © 2007 Springer Science+Business Media, LLC.
• Kennedy, T. (2006). Compact packings of the plane with two sizes of discs. Discrete and Computanional Geometry, 35(2), 255-267.
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  Abstract: We consider packings of the plane using discs of radius 1 and r. A packing is compact if every disc D is tangent to a sequence of discs D1, D2, ..., Dn such that Di is tangent to D i+1. We prove that there are only nine values of r with r < 1 for which such packings are possible. For each of the nine values we describe the possible compact packings. © 2005 Springer Science+Business Media, Inc.
• Kennedy, T. (2004). Conformal invariance and stochastic Loewner evolution predictions for the 2D self-avoiding walk - Monte Carlo tests. Journal of Statistical Physics, 114(1-2), 51-78.
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  Abstract: Simulations of the two-dimensional self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm, and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's stochastic Loewner evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance implies that if we map infinite length walks in the cut-plane into the half plane using the conformal map z → √z, then the resulting walks will have the same distribution as the SAW in the half plane. The simulations show excellent agreement between the distributions.
• Datta, N., & Kennedy, T. (2003). Instability of interfaces in the antiferromagnetic XXZ chain at zero temperature. Communications in Mathematical Physics, 236(3), 477-511.
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  Abstract: For the antiferromagnetic, highly anisotropic XZ and XXZ quantum spin chains, we impose periodic boundary conditions on chains with an odd number of sites to force an interface (or kink) into the chain. We prove that the energy of the interface depends on the momentum of the state. This shows that at zero temperature the interface in such chains is not stable. This is in contrast to the ferromagnetic XXZ chain for which the existence of localized interface ground states has been proven for any amount of anisotropy in the Ising-like regime.
• Datta, N., & Kennedy, T. (2002). Expansions for one quasiparticle states in spin 1/2 systems. Journal of Statistical Physics, 108(3-4), 373-399.
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  Abstract: Convergent expansions of the wavefunctions for the ground state and low-lying excited states of quantum transverse Ising systems are obtained. These expansions are employed to prove that the dispersion relation for a single quasi-particle has a convergent expansion.
• Kennedy, T. (2002). A faster implementation of the pivot algorithm for self-avoiding walks. Journal of Statistical Physics, 106(3-4), 407-429.
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  Abstract: The pivot algorithm is a Markov Chain Monte Carlo algorithm for simulating the self-avoiding walk. At each iteration a pivot which produces a global change in the walk is proposed. If the resulting walk is self-avoiding, the new walk is accepted; otherwise, it is rejected. Past implementations of the algorithm required a time O(N) per accepted pivot, where N is the number of steps in the walk. We show how to implement the algorithm so that the time required per accepted pivot is O(Nq) with q < 1. We estimate that q is less than 0.57 in two dimensions, and less than 0.85 in three dimensions. Corrections to the O(Nq) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(Nq) cannot be seen for walk lengths which can be simulated. In simulations the effective q is around 0.7 in two dimensions and 0.9 in three dimensions. Comparisons with simulations that use the standard implementation of the pivot algorithm using a hash table indicate that our implementation is faster by as much as a factor of 80 in two dimensions and as much as a factor of 7 in three dimensions. Our method does not require the use of a hash table and should also be applicable to the pivot algorithm for off-lattice models.
• Kennedy, T. (2002). Monte Carlo tests of stochastic Loewner evolution predictions for the 2D self-avoiding walk. Physical Review Letters, 88(13), 1306011-1306014.
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  PMID: 11955086;Abstract: Monte Carlo tests of stochastic Loewner evolution (SLE) predictions were carried out for a two-dimensional (2D) self-avoiding walk (SAW). The SAW was simulated using the pivot algorithm. The simulations of the SAW walk in a half plane showed that the distributions of two particular random variables related to the walk agreed well with the exact distributions of SLE8/3 for the random variables. This supported the conjuncture that the scaling limit of the SAW was SLE8/3.
• Haller, K., & Kennedy, T. (2001). Periodic ground states in the neutral Falicov-Kimball model in two dimensions. Journal of Statistical Physics, 102(1-2), 15-34.
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  Abstract: We consider the Falicov Kimball model in two dimensions in the neutral case, i.e., the number of mobile electrons is equal to the number of ions. For rational densities between 1/3 and 2/5 we prove that the ground state is periodic if the strength of the attraction between the ions and electrons is large enough. The periodic ground state is given by taking the one dimensional periodic ground state found by Lemberger and then extending it into two dimensions in such a way that the configuration is constant along lines at a 45 degree angle to the lattice directions.
• Kennedy, T. (1998). Phase separation in the neutral Falicov-Kimball model. Journal of Statistical Physics, 91(5-6), 829-843.
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  Abstract: The Falicov-Kimball model consists of spinless electrons and classical particles (ions) on a lattice. The electrons hop between nearest neighbor sites, while the ions do not. We consider the model with equal numbers of ions and electrons and with a large on-site attractive force between ions and electrons. For densities 1/4 and 1/5, the ion configuration in the ground state had been proved to be periodic. We prove that for density 2/9 it is periodic as well. However, for densities between 1/4 and 1/5 other than 2/9 we prove that the ion configuration in the ground state is not periodic. Instead there is phase separation. For densities in (1/5, 2/9) the ground-state ion configuration is a mixture of the density 1/5 and 2/9 ground-state ion configurations. For the interval (2/9, 1/4) it is a mixture of the density 2/9 and 1/4 ground states.
• Kennedy, T. (1997). Majority rule at low temperatures on the square and triangular lattices. Journal of Statistical Physics, 86(5-6), 1089-1107.
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  Abstract: We consider the majority-rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use the methods of van Enter, Fernández, and Sokal, who proved the renormalized measure is not Gibbsian for 7 by 7 blocks if the temperature is too low. For the triangular lattice we prove that a zero-temperature majority-rule transformation may be defined. The resulting renormalized Hamiltonian is local with 14 different types of interactions.
• Haller, K., & Kennedy, T. (1996). Absence of renormalization group pathologies near the critical temperature. Two examples. Journal of Statistical Physics, 85(5-6), 607-637.
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  Abstract: We consider real-space renormalization group transformations for Ising-type systems which are formally defined by exp[ - H′(σ′)] = ∑ T(σ, σ′) exp[ - H(σ)] where T(σ, σ′) is a probability kernel, i.e., ∑σ′, T(σ, σ′) = 1, for every configuration σ. For each choice of the block spin configuration σ′, let μσ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional to T(σ, σ′) exp[ - H(σ)]. We give a condition which is sufficient to imply that the renormalized Hamiltonian H′ is defined. Roughly speaking, the condition is that the collection of measures μσ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacing b = 2 on the square lattice with β < 1.36βc and the Kadanoff transformation with parameter p on the triangular lattice in a subset of the β, p plane that includes values of β greater than βc.
• Kennedy, T. (1994). Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty. Journal of Statistical Physics, 77(3-4), 565-579.
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  Abstract: We consider a weakly self-avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp[-β|t-s|-p] where t and s are the times at which the common site is visited and p is a parameter. We prove that if p3/2, then the walk should have diffusive behavior. The proof and the heuristic argument make use of a real-space renormalization group transformation. © 1994 Plenum Publishing Corporation.
• Kennedy, T. (1994). Non-positive matrix elements for Hamiltonians of spin-1 chains. Journal of Physics Condensed Matter, 6(39), 8015-8022.
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  Abstract: For a large class of one-dimensional spin-1 Hamiltonians with open boundary conditions, we show that there is a unitary transformation for which the off-diagonal matrix elements of the transformed Hamiltonian are non-positive. We use this to show that the ground state of a finite chain is at most fourfold degenerate, and that the expectation of the string observable of den Nijs and Rommelse in the ground state is bounded below by the expectation of the usual Neel order parameter. (This was proved for a smaller class of Hamiltonians by Kennedy and Tasaki.) The class of Hamiltonians to which our results apply include the general isotropic Hamiltonian Σi[Si · Si+1 - β(Si · Si+1)2] for β > -1. For the usual Heisenberg Hamiltonian the transformed Hamiltonian is - ΣiTi · Ti+1 where the operators T = (Tx, Ty, Tz) satisfy anticommutation relations like {Tx, Ty} = Tz. We can also use this transformation to obtain variational bounds on the ground-state energy. The transformation used here is closely related to the unitary operator introduced by Kennedy and Tasaki.
• Kennedy, T. (1994). Some rigorous results on the ground states of the Falicov-Kimball model. Reviews in Mathematical Physics, 06(05a), 901-925. doi:10.1142/s0129055x94000298
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  The Falicov-Kimball model consists of nuclei which are not allowed to hop and spinless electrons which can hop between nearest neighbor sites. There is an on-site interaction between electrons and nuclei. We consider the model in two dimensions with a large attractive potential. For the neutral model with densities between 1/4 and 1/2 we prove that the configuration of the nuclei in the ground state must consist of parallel lines of lattice sites which are either completely occupied by nuclei or completely free of nuclei. (The angle of the lines with respect to the lattice depends on the density. Some mild assumptions on the ground state are needed for this result.) For densities 1/3, 1/4 and 1/5 we prove that the ground state configuration of the nuclei is indeed that which had been conjectured [8]. For the nonneutral model we show that if the model is close to neutrality in the sense that both the electron and nuclear densities are close to 1/2, 1/3, 1/4 or 1/5 then the configuration of the nuclei in the ground state is close to the nuclear ground state for the neutral model with density 1/2, 1/3, 1/4 or 1/5.
• Kennedy, T. (1993). Some rigorous results on majority rule renormalization group transformations near the critical point. Journal of Statistical Physics, 72(1-2), 15-37.
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  Abstract: We consider the majority rule renormalization group transformation with two-by-two blocks for the Ising model on a two-dimensional square lattice. For three particular choices of the block spin configuration we prove that the model conditioned on the block spin configuration remains in the high-temperature phase even when the temperature is slightly below the critical temperature of the ordinary Ising model with no conditioning. We take as the definition of the infinite-volume limit an equation introduced in earlier work by the author. We use a computer to find an approximate solution of this equation and verify a condition which implies the existence of an exact solution. © 1993 Plenum Publishing Corporation.
• Kennedy, T. (1992). Solutions of the Yang-Baxter equation for isotropic quantum spin chains. Journal of Physics A: Mathematical and General, 25(10), 2809-2817.
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  Abstract: We consider solutions of the Yang-Baner quation such that the logarithmic derivative of the transfer matrix yields a quantum spin Hamiltonian which is isotropic in spin space, i.e. SU(2)-invariant. Four such solutions are known far each value of the spin S. (For S = 1/2 they degenerate into the same solution, and for S = 1 they only give three different solutions.) For S ≤ 6 we show that these are the only solutions which are SU(2)-invariant, except for S = 3 when there is a fifth solution. ©1992 IOP Publishing Ltd.
• Kennedy, T., & Tasaki, H. (1992). Hidden Z2xZ2 symmetry breaking in Haldane-gap antiferromagnets. Physical Review B, 45(1), 304-307.
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  Abstract: We show that the Haldane phase of the S=1 antiferromagnetic chain is closely related to the breaking of a hidden Z2×Z2 symmetry. When the chain is in the Haldane phase, this Z2×Z2 symmetry is fully broken, but when the chain is in a massive phase other than the Haldane phase, e.g., the Ising phase or the dimerized phase, this symmetry is broken only partially or not at all. The hidden Z2×Z2 symmetry is revealed by introducing a nonlocal unitary transformation of the chain. This unitary transformation also leads to a simple variational calculation which qualitatively reproduces the phase diagram of the S=1 chain. © 1992 The American Physical Society.
• Kennedy, T., & Tasaki, H. (1992). Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains. Communications in Mathematical Physics, 147(3), 431-484.
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  Abstract: We study the phase diagram of S=1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of a Z2×Z2 symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Néel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of "diagonally dominant" Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hidden Z2×Z2 symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models. © 1992 Springer-Verlag.
• Guo, D., Kennedy, T., & Mazumdar, S. (1991). Spin-Peierls transitions in S > 1 2 Heisenberg chains. Synthetic Metals, 43(1-2), 3513-.
• Kennedy, T. (1991). Ornstein-Zernike decay in the ground state of the quantum Ising model in a strong transverse field. Communications in Mathematical Physics, 137(3), 599-615.
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  Abstract: We consider the quantum mechanical Ising ferromagnet in a strong transverse magnetic field in nay number of dimensions, d. We prove that in the ground state the power law correction to the exponential decay of the two point function is d/2. The proof begins by writing the ground state as a classical system in one more dimension. (Thus the classical Ornstein-Zernike power of (d-1)/2 becomes d/2). We then develop a convergent polymer expansion and use the techniques of Bricmont and Fröhlich [5]. © 1991 Springer-Verlag.
• Guo, D., Kennedy, T., & Mazumdar, S. (1990). Spin-Peierls transitions in S>1/2 Heisenberg chains. Physical Review B, 41(13), 9592-9595.
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  Abstract: Exact diagonalizations are done to calculate the energy gain from bond alternation in finite spin chains as a function of chain length. Calculations are done for S=1/2, 1, 3/2, and 2. Qualitative differences are found between half-integer and interger spins. Our results (i) indicate the absence of the spin-Peierls transition in systems with integer spins and (ii) suggest a transition in systems with half-integer spins. © 1990 The American Physical Society.
• Kennedy, T. (1990). A fixed-point equation for the high-temperature phase of discrete lattice spin systems. Journal of Statistical Physics, 59(1-2), 195-220.
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  Abstract: A fixed-point equation on an infinite-dimensional space is proposed as an alternative to the usual definition of the infinite-volume limit in discrete lattice spin systems in the high-temperature phase. It is argued heuristically that the free energy and correlation functions one obtains by solving this equation agree with the usual definitions of these quantities. A theorem is then proved that says that if a certain finite-volume condition is satisfied, then this fixed-point equation has a solution and the resulting free energy is analytic in the parameters in the Hamiltonian. For particular values of the temperature this finite-volume condition may be checked with the help of a computer. The two-dimensional Ising model is considered as a test case, and it is shown that the finite-volume condition is satisfied for β≤0.77βcritical. © 1990 Plenum Publishing Corporation.
• Kennedy, T. (1990). Exact diagonalisations of open spin-1 chains. Journal of Physics: Condensed Matter, 2(26), 5737-5745.
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  Abstract: The author numerically computes the two lowest eigenvalues of finite length spin-1 chains with the Hamiltonian H= Sigma i(Si.S i+1- beta (Si.Si+1)2) and open boundary conditions. For a range of beta , including the value 0, he finds that the difference of the two eigenvalues decays exponentially with the length of the chain. This exponential decay provides further evidence that these spin chains are in a massive phase as first predicted by Haldane (1982). The correlation length xi of the chain can be estimated using this exponential decay. He finds estimates of xi for the Heisenberg chain ( beta =0) that range from 6.7 to 7.8 depending on how one extrapolates to infinite length.
• Affleck, I., Kennedy, T., Lieb, E. H., & Tasaki, H. (1988). Valence bond ground states in isotropic quantum antiferromagnets. Communications in Mathematical Physics, 115(3), 477-528.
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  Abstract: Haldane predicted that the isotropic quantum Heisenberg spin chain is in a "massive" phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds. © 1988 Springer-Verlag.
• Kennedy, T., Lieb, E. H., & Shastry, B. (1988). Existence of Néel order in some spin-1/2 Heisenberg antiferromagnets. Journal of Statistical Physics, 53(5-6), 1019-1030.
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  Abstract: The methods of Dyson, Lieb, and Simon are extended to prove the existence of Néel order in the ground state of the 3D spin-1/2 Heisenberg antiferromagnet on the cubic lattice. We also consider the spin-1/2 antiferromagnet on the cubic lattice with the coupling in one of the three lattice directions taken to be r times its value in the other two lattice directions. We prove the existence of Néel order for 0.16≤r≤1. For the 2D spin-1/2 model we give a series of inequalities which involve the two-point function only at short distances and each of which would by itself imply Néel order. © 1988 Plenum Publishing Corporation.
• Kennedy, T., Lieb, E. H., & Shastry, B. S. (1988). The XY model has long-range order for all spins and all dimensions greater than one. Physical Review Letters, 61(22), 2582-2584.
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  Abstract: The quantum XY model of interacting spins on a hypercubic lattice has long-range order in the ground state for all values of the spin and all dimensions greater than one. We also show that in the limit of high dimension the spontaneous magnetization converges to the spontaneous magnetization of the Néel state. © 1988 The American Physical Society.
• Kennedy, T., Lieb, E. H., & Tasaki, H. (1988). A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state. Journal of Statistical Physics, 53(1-2), 383-415.
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  Abstract: We continue the study of valence-bond solid antiferromagnetic quantum Hamiltonians. These Hamiltonians are invariant under rotations in spin space. We prove that a particular two-dimensional model from this class (the spin-3/2 model on the hexagonal lattice) has a unique ground state in the infinite-volume limit and hence no Néel order. Moreover, all truncated correlation functions decay exponentially in this ground state. We also characterize all the finite-volume ground states of these models (in every dimension), and prove that the two-point correlation function of the spin-2 square lattice model with periodic boundary conditions has exponential decay. © 1988 Plenum Publishing Corporation.
• Affleck, I., Kennedy, T., Lieb, E. H., & Tasaki, H. (1987). Rigorous results on valence-bond ground states in antiferromagnets. Physical Review Letters, 59(7), 799-802.
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  Abstract: We present rigorous results on a phase in antiferromagnets in one dimension and more, which we call a valence-bond solid. The ground state is simply constructed out of valence bonds, is nondegenerate, and breaks no symmetries. There is an energy gap and an exponentially decaying correlation function. Physical applications are mentioned. © 1987 The American Physical Society.
• Brydges, D. C., & Kennedy, T. (1987). Mayer expansions and the Hamilton-Jacobi equation. Journal of Statistical Physics, 48(1-2), 19-49.
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  Abstract: We review the derivation of Wilson's differential equation in (infinitely) many variables, which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal "integration out" is performed. We show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. The differential equation approach gives a remarkable identity for "connected parts" and precise estimates which include criteria for convergence of iterated Mayer expansions. Applications include the Yukawa gas in two dimensions past the Β=4 π threshold and another derivation of some earlier results of Göpfert and Mack. © 1987 Plenum Publishing Corporation.
• Kennedy, T., & Lieb, E. H. (1987). Proof of the Peierls instability in one dimension. Physical Review Letters, 59(12), 1309-1312.
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  Abstract: Fröhlich and Peierls showed that a one-dimensional system with a half-filled band can lower its ground-state energy by a dimerization from period 1 to period 2. It was an open question whether or not this dimerization was exact, i.e., whether additional symmetry breaking would further lower the energy. We prove that the dimerization is exact for a periodic chain of infinitely massive, harmonically bound atoms with nearest-neighbor electron hopping matrix elements that vary linearly with the nearest-neighbor distance. © 1987 The American Physical Society.
• Kennedy, T., & King, C. (1986). Spontaneous symmetry breakdown in the abelian Higgs model. Communications in Mathematical Physics, 104(2), 327-347.
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  Abstract: For the abelian Higgs model we introduce a new gauge invariant observable which in Landau gauge is {Mathematical expression}. In three or more dimensions this observable is used to show that the global gauge symmetry is spontaneously broken in the lattice theory for a suitable range of parameters. This observable also provides a gauge invariant order parameter for the phase transition in this model. © 1986 Springer-Verlag.
• Kennedy, T., & Lieb, E. H. (1986). An itinerant electron model with crystalline or magnetic long range order. Physica A: Statistical Mechanics and its Applications, 138(1-2), 320-358.
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  Abstract: A quantum mechanical lattice model of fermionic electrons interacting with infinitely massive nuclei is considered. (It can be viewed as a modified Hubbard model in which the spin-up electrons are not allowed to hop.) The electron-nucleus potential is "on-site" only. Neither this potential alone nor the kinetic energy alone can produce long range order. Thus, if long range order exists in this model it must come from an exchange mechanism. N, the electron plus nucleus number, is taken to be less than or equal to the number of lattice sites. We prove the following: (i) For all dimensions, d, the ground state has long range order; in fact it is a perfect crystal with spacing √2 times the lattice spacing. A gap in the ground state energy always exists at the half-filled band point (N = number of lattice sites). (ii) For small, positive temperature, T, the ordering persists when d ≥ 2. If T is large there is no long range order and there is exponential clustering of all correlation functions. © 1986.
• Federbush, P., & Kennedy, T. (1985). Surface effects in Debye screening. Communications in Mathematical Physics, 102(3), 361-423.
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  Abstract: A thermodynamic system of equally charged, plus and minus, classical particles constrained to move in a (spherical) ball is studied in a region of parameters in which Debye screening takes place. The activities of the two charge species are not taken as necessarily equal. We must deal with two physically interesting surface effects, the formation of a surface charge layer, and long range forces reaching around the outside of the spherical volume. This is an example in as much as 1) general charge species are not considered, 2) the volume is taken as a ball, 3) a simple choice for the short range forces (necessary for stability) is taken. We feel the present system is general enough to exhibit all the interesting physical phenomena, and that the methods used are capable of extension to much more general systems. The techniques herein involve use of the sine-Gordon transformation to get a continuum field problem which in turn is studied via a multi-phase cluster expansion. This route follows other recent rigorous treatments of Debye screening. © 1985 Springer-Verlag.
• Kennedy, T. (1985). Long range order in the anisotropic quantum ferromagnetic Heisenberg model. Communications in Mathematical Physics, 100(3), 447-462.
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  Abstract: We study the anisotropic quantum mechanical ferromagnetic Heisenberg model. By anisotropic we mean that the x and y exchange constants are equal but smaller than the z exchange constant. We show that for any amount of anisotropy there is long range order in two or more dimensions at low enough temperature. We also develop a convergent low temperature expansion and use it to prove exponential decay of the truncated correlation functions. © 1985 Springer-Verlag.
• Kennedy, T., & King, C. (1985). Symmetry Breaking in the Lattice Abelian Higgs Model. Physical Review Letters, 55(8), 776-778.
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  Abstract: A new gauge-invariant order parameter is introduced for the Abelian Higgs model and used to prove the existence of a phase transition for the lattice theory in three or more dimensions. In Landau gauge this order parameter is the limit of (x)»(y) as x-y. © 1985 The American Physical Society.
• Kennedy, T. (1984). Mean field theory for Coulomb systems. Journal of Statistical Physics, 37(5-6), 529-559.
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  Abstract: We study a classical charge symmetric system with an external charge distribution q in three dimensions in the limit that the plasma parameter e{open}→ zero. We prove that if q is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field Ψ(x) where Ψ is given by-ΔΨ+ 2z sinh(βΨ) =q This is the mean field equation of Debye and Hückel. The proof uses the sine-Gordon transformation, the Mayer expansion, and a correlation inequality. © 1984 Plenum Publishing Corporation.
• Kennedy, T. (1983). Debye-Hückel theory for charge symmetric Coulomb systems. Communications in Mathematical Physics, 92(2), 269-294.
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  Abstract: It is proven that the pressure, density and correlation functions of a classical charge symmetric Coulomb system are asymptotic as the plasma parameter ε tends to zero to the approximations predicted by the Debye-Hückel theory. These approximations consist of the ideal gas term plus a term of one lower order in ε. The sine-Gordon transformation and some new correlation inequalities for the associated functional integrals are used. © 1983 Springer-Verlag.
• Kennedy, T. (1982). A lower bound on the partition function for a classical charge symmetric system. Journal of Statistical Physics, 28(4), 633-638.
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  Abstract: A lower bound is obtained for the grand canonical partition function (and hence for the pressure) of a charge symmetric system with positive definite interaction. For the Coulomb interaction the lower bound on the pressure is the Debye-Hückel approximation. © 1982 Plenum Publishing Corporation.