arXiv subjects

Tom Kennedy

Publications and source records attributed to Tom Kennedy.

At least 19 recordsLinked to original sources

arXiv2025

Transformational astrophysics and exoplanet science with Habitable Worlds Observatory's High Resolution Imager

Habitable Worlds Observatory (HWO) will be NASA's flagship space telescope of the 2040s, designed to search for life on other planets and to transform broad areas of astrophysics. NASA are seeking international partners, and the UK is well-placed to lead the design and construction of its imaging camera - which is likely to produce the mission's most visible public impact. Early participation in the mission would return investment to UK industry, and bring generational leadership for the UK in space science, space technology, and astrophysics.

astro-ph.IM↗

arXiv2025

Tensor Renormalization Group Meets Computer Assistance

Tensor renormalization group, originally devised as a numerical technique, is emerging as a rigorous analytical framework for studying lattice models in statistical physics. Here we introduce a new renormalization map - the 2x1 map - which coarse-grains the lattice anisotropically by a factor of two in one direction followed by a 90-degree rotation. We develop a novel graphical language that translates the action of the 2x1 map into a system of inequalities on tensor components, with rigorous estimates in the Hilbert-Schmidt norm. We define a finite-dimensional "bounding box" called the hat-tensor, and a master function governing its RG flow. Iterating this function numerically, we establish convergence to the high-temperature fixed point for tensors lying within a quantifiable neighborhood. Our main theorem shows that tensors with deviations bounded by 0.02 in 63 orthogonal sectors flow to the fixed point. We also apply the method to specific models - the 2D Ising and XY models - obtaining explicit bounds on their high-temperature phase. This work brings the Tensor RG program closer towards a rigorous, computer-assisted construction of critical fixed points.

cond-mat.stat-mech↗

arXiv2024

Transfer Matrix and Lattice Dilatation Operator for High-Quality Fixed Points in Tensor Network Renormalization Group

Tensor network renormalization group maps study critical points of 2d lattice models like the Ising model by finding the fixed point of the RG map. In a prior work arXiv:2408.10312 we showed that by adding a rotation to the RG map, the Newton method could be implemented to find an extremely accurate fixed point. For a particular RG map (Gilt-TNR) we studied the spectrum of the Jacobian of the RG map at the fixed point and found good agreement between the eigenvalues corresponding to relevant and marginal operators and their known exact values. In this companion work we use two further methods to extract many more scaling dimensions from this Newton method fixed point, and compare the numerical results with the predictions of conformal field theory (CFT). The first method is the well-known transfer matrix (TM), while the second method we refer to as the lattice dilatation operator (LDO). We introduce some extensions of these method that provide also spins of the CFT operators, modulo an integer. With comparable computing resources, the TM and LDO methods perform equally well. The agreement for the scaling dimensions and spins is excellent up to the scaling dimesnion 4.125, and reasonably good up to 2 units higher. Some of the eigenvalues of the Jacobian of the RG map can come from perturbations associated with total derivative interactions and so are not universal. In some past studies (arXiv:2102.08136, arXiv:2305.09899) such non-universal eigenvalues did not appear in the Jacobian. We explain this surprising result by showing that their RG map has the unusual property that the Jacobian is equivalent to the LDO operator.

cond-mat.stat-mech↗

arXiv2024

Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group

In the tensor network approach to statistical physics, properties of the critical point of a 2D lattice model are encoded by a four-legged tensor which is a fixed point of an RG map. The traditional way to find the fixed point tensor consists in iterating the RG map after having tuned the temperature to criticality. Here we develop a different and more direct technique, which solves the fixed point equation via the Newton method. This is challenging due to the existence of marginal deformations -- linear transformations of the coordinate frame, which parametrize a two-dimensional family of fixed points. We address this challenge by including a 90 degree rotation into the RG map. This flips the sign of the problematic marginal eigenvalues, rendering the fixed point isolated and accessible via the Newton method. We demonstrate the power of this technique via explicit computations for the 2D Ising and 3-state Potts models. Using the Gilt-TNR algorithm at bond dimension $\chi=30$, we find the fixed point tensors with $10^{-9}$ accuracy, much higher than what was previously achieved.

cond-mat.stat-mech↗

arXiv2022

Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point

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.

math-ph↗

arXiv2021

Tensor RG approach to high-temperature fixed point

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.

math-ph↗

arXiv2020

Absence of renormalization group pathologies in some critical Dyson-Ising ferromagnets

The Dyson-Ising ferromagnet is a one-dimensional Ising model with a power law interaction. When the power is between -1 and -2, the model has a phase transition. Van Enter and Le Ny proved that at sufficiently low temperatures the decimation renormalization group transformation is not defined in the sense that the renormalized measure is not a Gibbs measure. We consider a modified model in which the nearest neighbor couplings are much larger than the other couplings. For a family of Hamiltonians which includes critical cases, we prove that the first step of the renormalization group transformation can be rigorously defined for majority rule and decimation.

math-ph↗

arXiv2018

Conformal invariance of the loop-erased percolation explorer

We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes within one lattice spacing of itself. We define a new curve by erasing these near-loops as we traverse the interface. Our Monte Carlo simulations of this model lead us to conclude that the scaling limit of this loop-erased percolation interface is conformally invariant and has fractal dimension 4/3. However, it is not SLE_8/3. We also consider the process in which a near-loop is when the explorer comes within two lattice spacings of itself.

math.PR↗

arXiv2018

The Large Area Detector onboard the eXTP mission

The eXTP (enhanced X-ray Timing and Polarimetry) mission is a major project of the Chinese Academy of Sciences (CAS) and China National Space Administration (CNSA) currently performing an extended phase A study and proposed for a launch by 2025 in a low-earth orbit. The eXTP scientific payload envisages a suite of instruments (Spectroscopy Focusing Array, Polarimetry Focusing Array, Large Area Detector and Wide Field Monitor) offering unprecedented simultaneous wide-band X-ray spectral, timing and polarimetry sensitivity. A large European consortium is contributing to the eXTP study and it is expected to provide key hardware elements, including a Large Area Detector (LAD). The LAD instrument for eXTP is based on the design originally proposed for the LOFT mission within the ESA context. The eXTP/LAD envisages a deployed 3.4 m2 effective area in the 2-30 keV energy range, achieved through the technology of the large-area Silicon Drift Detectors - offering a spectral resolution of up to 200 eV FWHM at 6 keV - and of capillary plate collimators - limiting the field of view to about 1 degree. In this paper we provide an overview of the LAD instrument design, including new elements with respect to the earlier LOFT configuration.

astro-ph.IM↗

arXiv2018

A non-intersecting random walk on the Manhattan lattice and SLE_6

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 SLE$_6$. We test this conjecture with Monte Carlo simulations of the random walk model and find strong support for the conjecture.

math.PR↗

arXiv2016

The first order correction to the exit distribution for some random walks

We study three different random walk models on several two-dimensional lattices by Monte Carlo simulations. One is the usual nearest neighbor random walk. Another is the nearest neighbor random walk which is not allowed to backtrack. The final model is the smart kinetic walk. For all three of these models the distribution of the point where the walk exits a simply connected domain $D$ in the plane converges weakly to harmonic measure on $\partial D$ as the lattice spacing $δ\rightarrow 0$. Let $ω(0,|dz|;D)$ be harmonic measure for $D$, and let $ω_δ(0,|dz|;D)$ be the discrete harmonic measure for one of the random walk models. Our definition of the random walk models is unusual in that we average over the orientation of the lattice with respect to the domain. We are interested in the limit of $(ω_δ(0,|dz|;D)- ω(0,|dz|;D))/δ$. Our Monte Carlo simulations of the three models lead to the conjecture that this limit equals $c_{M,L} \, ρ_D(z) |dz|$, where the function $ρ_D(z)$ depends on the domain, but not on the model or lattice, and the constant $c_{M,L}$ depends on the model and on the lattice, but not on the domain. So there is a form of universality for this first order correction. We also give an explicit formula for the conjectured density $ρ_D$.

math.PR↗

arXiv2016

The difference between a discrete and continuous harmonic measure

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius of $h$. For a simply connected domain $D$ in the plane, let $ω_h(0,\cdot;D)$ be the discrete harmonic measure at $0\in D$ associated with this random walk, and $ω(0,\cdot;D)$ be the (continuous) harmonic measure at $0$. For domains $D$ with analytic boundary, we prove there is a bounded continuous function $σ_D(z)$ on $\partial D$ such that for functions $g$ which are in $C^{2+α}(\partial D)$ for some $α>0$ $$ \lim_{h\downarrow 0} \frac{\int_{\partial D} g(ξ) ω_h(0,|dξ|;D) -\int_{\partial D} g(ξ)ω(0,|dξ|;D)}{h} = \int_{\partial D}g(z) σ_D(z) |dz|. $$ We give an explicit formula for $σ_D$ in terms of the conformal map from $D$ to the unit disc. The proof relies on some fine approximations of the potential kernel and Green's function of the random walk by their continuous counterparts, which may be of independent interest.

math.PR↗

arXiv2015

The Smart Kinetic Self-Avoiding Walk and Schramm-Loewner Evolution

The smart kinetic self-avoiding walk (SKSAW) is a random walk which never intersects itself and grows forever when run in the full-plane. At each time step the walk chooses the next step uniformly from among the allowable nearest neighbors of the current endpoint of the walk. In the full-plane a nearest neighbor is allowable if it has not been visited before and there is a path from the nearest neighbor to infinity through sites that have not been visited before. It is well known that on the hexagonal lattice the SKSAW in a bounded domain between two boundary points is equivalent to an interface in critical percolation, and hence its scaling limit is the chordal Schramm-Loewner evolution with kappa=6 (SLE_6). Like SLE there are variants of the SKSAW depending on the domain and the initial and terminal points. On the hexagonal lattice these variants have been shown to converge to the corresponding version of SLE_6. We conjecture that the scaling limit of all these variants on any regular lattice is the corresponding version of SLE_6. We test this conjecture for the square lattice by simulating the SKSAW in the full-plane and find excellent agreement with the predictions of full-plane SLE_6.

math.PR↗

arXiv2014

Conformal invariance predictions for the three-dimensional self-avoiding walk

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. The ensembles of SAW's used to define these hitting densities involve walks of arbitrary lengths, and so these ensembles cannot be directly studied by the pivot Monte Carlo algorithm for the SAW. We show that these mixed length ensembles should have the same scaling limit as certain weighted ensembles that only involve walks with a single length, thus providing a fast method for simulating these ensembles. Preliminary simulations which found good agreement between the predictions and Monte Carlo simulations for the SAW were reported in [14]. In this paper we present more accurate simulations testing the predictions and find even stronger support for the prediction that the SAW is conformally invariant in three dimensions.

math-ph↗

arXiv2013

Conformal invariance of the 3D self-avoiding walk

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.

math-ph↗

arXiv2012

Self-avoiding walks in a rectangle

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 ratio of probabilities 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 \cite{BLWW04}. Here we consider instead the more difficult problem of a self-avoiding walk in the scaling limit, and pose the same question. Assuming that the scaling limit of SAW is conformally invariant, we evaluate, asymptotically, the same ratio of probabilities. For the SAW case we find the probability ratio is approximately 200 times greater than for Brownian motion.

math-ph↗

arXiv2011

Simulating self-avoiding walks in bounded domains

Let D be a domain in the plane containing the origin. We are interested in the ensemble of self-avoiding walks (SAW's) in D which start at the origin and end on the boundary of the domain. We introduce an ensemble of SAW's 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 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.

math.PR↗

arXiv2011

Lattice effects in the scaling limit of the two-dimensional self-avoiding walk

We consider the two-dimensional self-avoiding walk (SAW) in a simply connected domain that contains the origin. The SAW starts at the origin and ends somewhere on the boundary. The distribution of the endpoint along the boundary is expected to differ from the SLE partition function prediction for this distribution because of lattice effects that persist in the scaling limit. We give a precise conjecture for how to compute this lattice effect correction and support our conjecture with simulations. We also give a precise conjecture for the lattice corrections that persist in the scaling limit of the lambda-SAW walk.

math.PR↗