TY - JOUR AB - Let P be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty n-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that P is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the k-matching-free process, where P is the property of not containing a matching of size k. We are able to analyse the behaviour of this process for a wide range of values of k; in particular we prove that if k=o(n) or if n−2k=o(n−−√/logn) then this process is likely to terminate in a k-matching-free graph with the maximum possible number of edges, as characterised by Erdős and Gallai. We also show that these bounds on k are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime. AU - Krivelevich, Michael AU - Kwan, Matthew Alan AU - Loh, Po‐Shen AU - Sudakov, Benny ID - 9567 IS - 4 JF - Random Structures and Algorithms SN - 1042-9832 TI - The random k‐matching‐free process VL - 53 ER - TY - JOUR AB - Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if p≥(logn+ω(1))/n then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically n!(p(1+o(1)))n. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically n!(logn/n(1+o(1)))n directed Hamilton cycles. AU - Ferber, Asaf AU - Kwan, Matthew Alan AU - Sudakov, Benny ID - 9565 IS - 4 JF - Random Structures and Algorithms SN - 1042-9832 TI - Counting Hamilton cycles in sparse random directed graphs VL - 53 ER - TY - JOUR AB - An intercalate in a Latin square is a 2×2 Latin subsquare. Let N be the number of intercalates in a uniformly random n×n Latin square. We prove that asymptotically almost surely N≥(1−o(1))n2/4, and that EN≤(1+o(1))n2/2 (therefore asymptotically almost surely N≤fn2 for any f→∞). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in two fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low-discrepancy Latin squares. AU - Kwan, Matthew Alan AU - Sudakov, Benny ID - 9568 IS - 2 JF - Random Structures and Algorithms SN - 1042-9832 TI - Intercalates and discrepancy in random Latin squares VL - 52 ER - TY - JOUR AB - We say a family of sets is intersecting if any two of its sets intersect, and we say it is trivially intersecting if there is an element which appears in every set of the family. In this paper we study the maximum size of a non-trivially intersecting family in a natural “multi-part” setting. Here the ground set is divided into parts, and one considers families of sets whose intersection with each part is of a prescribed size. Our work is motivated by classical results in the single-part setting due to Erdős, Ko and Rado, and Hilton and Milner, and by a theorem of Frankl concerning intersecting families in this multi-part setting. In the case where the part sizes are sufficiently large we determine the maximum size of a non-trivially intersecting multi-part family, disproving a conjecture of Alon and Katona. AU - Kwan, Matthew Alan AU - Sudakov, Benny AU - Vieira, Pedro ID - 9587 JF - Journal of Combinatorial Theory Series A SN - 0097-3165 TI - Non-trivially intersecting multi-part families VL - 156 ER - TY - JOUR AB - We investigate the thermodynamics and kinetics of a hydrogen interstitial in magnetic α-iron, taking account of the quantum fluctuations of the proton as well as the anharmonicities of lattice vibrations and hydrogen hopping. We show that the diffusivity of hydrogen in the lattice of bcc iron deviates strongly from an Arrhenius behavior at and below room temperature. We compare a quantum transition state theory to explicit ring polymer molecular dynamics in the calculation of diffusivity. We then address the trapping of hydrogen by a vacancy as a prototype lattice defect. By a sequence of steps in a thought experiment, each involving a thermodynamic integration, we are able to separate out the binding free energy of a proton to a defect into harmonic and anharmonic, and classical and quantum contributions. We find that about 30% of a typical binding free energy of hydrogen to a lattice defect in iron is accounted for by finite temperature effects, and about half of these arise from quantum proton fluctuations. This has huge implications for the comparison between thermal desorption and permeation experiments and standard electronic structure theory. The implications are even greater for the interpretation of muon spin resonance experiments. AU - Cheng, Bingqing AU - Paxton, Anthony T. AU - Ceriotti, Michele ID - 9665 IS - 22 JF - Physical Review Letters SN - 0031-9007 TI - Hydrogen diffusion and trapping in α-iron: The role of quantum and anharmonic fluctuations VL - 120 ER - TY - JOUR AB - The curvature dependence of interfacial free energy, which is crucial in quantitatively predicting nucleation kinetics and the stability of bubbles and droplets, is quantified by the Tolman length δ. For solid-liquid interfaces, however, δ has never been computed directly due to various theoretical and practical challenges. Here we perform a direct evaluation of the Tolman length from atomistic simulations of a solid-liquid planar interface in out-of-equilibrium conditions, by first computing the surface tension from the amplitude of thermal capillary fluctuations of a localized version of the Gibbs dividing surface and by then calculating how much the surface energy changes when it is defined relative to the equimolar dividing surface. We computed δ for a model potential, and found a good agreement with the values indirectly inferred from nucleation simulations. The agreement not only validates our approach but also suggests that the nucleation free energy of the system can be perfectly described using classical nucleation theory if the Tolman length is taken into account. AU - Cheng, Bingqing AU - Ceriotti, Michele ID - 9659 IS - 23 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Communication: Computing the Tolman length for solid-liquid interfaces VL - 148 ER - TY - JOUR AB - We present a field-data rich modelling analysis to reconstruct the climatic forcing, glacier response, and runoff generation from a high-elevation catchment in central Chile over the period 2000–2015 to provide insights into the differing contributions of debris-covered and debris-free glaciers under current and future changing climatic conditions. Model simulations with the physically based glacio-hydrological model TOPKAPI-ETH reveal a period of neutral or slightly positive mass balance between 2000 and 2010, followed by a transition to increasingly large annual mass losses, associated with a recent mega drought. Mass losses commence earlier, and are more severe, for a heavily debris-covered glacier, most likely due to its strong dependence on snow avalanche accumulation, which has declined in recent years. Catchment runoff shows a marked decreasing trend over the study period, but with high interannual variability directly linked to winter snow accumulation, and high contribution from ice melt in dry periods and drought conditions. The study demonstrates the importance of incorporating local-scale processes such as snow avalanche accumulation and spatially variable debris thickness, in understanding the responses of different glacier types to climate change. We highlight the increased dependency of runoff from high Andean catchments on the diminishing resource of glacier ice during dry years. AU - Burger, Flavia AU - Ayala, Alvaro AU - Farias, David AU - Shaw, Thomas E. AU - MacDonell, Shelley AU - Brock, Ben AU - McPhee, James AU - Pellicciotti, Francesca ID - 12603 IS - 2 JF - Hydrological Processes KW - Water Science and Technology SN - 0885-6087 TI - Interannual variability in glacier contribution to runoff from a high‐elevation Andean catchment: Understanding the role of debris cover in glacier hydrology VL - 33 ER - TY - JOUR AB - Snow depth patterns over glaciers are controlled by precipitation, snow redistribution due to wind and avalanches, and the exchange of energy with the atmosphere that determines snow ablation. While many studies have advanced the understanding of ablation processes, less is known about winter snow patterns and their variability over glaciers. We analyze snow depth on Haut Glacier d'Arolla, Switzerland, in the two winter seasons 2006–2007 and 2010–2011 to (1) understand whether snow depth over an alpine glacier at the end of the accumulation season exhibits a behavior similar to the one observed on single slopes and vegetated areas; and (2) investigate the snow pattern consistency over the two accumulation seasons. We perform this analysis on a data set of high-resolution lidar-derived snow depth using variograms and fractal parameters. Our first main result is that snow depth patterns on the glacier exhibit a multiscale behavior, with a scale break around 20 m after which the fractal dimension increases, indicating more autocorrelated structure before the scale break than after. Second, this behavior is consistent over the two years, with fractal parameters and their spatial variability almost constant in the two seasons. We also show that snow depth patterns exhibit a distinct behavior in the glacier tongue and the upper catchment, with longer correlation distances on the tongue in the direction of the main winds, suggesting spatial distinctions that are likely induced by different processes and that should be taken into account when extrapolating snow depth from limited samples. AU - Clemenzi, I. AU - Pellicciotti, Francesca AU - Burlando, P. ID - 12605 IS - 10 JF - Water Resources Research KW - Water Science and Technology SN - 0043-1397 TI - Snow depth structure, fractal behavior, and interannual consistency over Haut Glacier d'Arolla, Switzerland VL - 54 ER - TY - JOUR AB - Glaciers in the high mountains of Asia provide an important water resource for millions of people. Many of these glaciers are partially covered by rocky debris, which protects the ice from solar radiation and warm air. However, studies have found that the surface of these debris-covered glaciers is actually lowering as fast as glaciers without debris. Water ponded on the surface of the glaciers may be partially responsible, as water can absorb atmospheric energy very efficiently. However, the overall effect of these ponds has not been thoroughly assessed yet. We study a valley in Nepal for which we have extensive weather measurements, and we use a numerical model to calculate the energy absorbed by ponds on the surface of the glaciers over 6 months. As we have not observed each individual pond thoroughly, we run the model 5,000 times with different setups. We find that ponds are extremely important for glacier melt and absorb energy 14 times as quickly as the debris-covered ice. Although the ponds account for 1% of the glacier area covered by rocks, and only 0.3% of the total glacier area, they absorb enough energy to account for one eighth of the whole valley's ice loss. AU - Miles, Evan S. AU - Willis, Ian AU - Buri, Pascal AU - Steiner, Jakob F. AU - Arnold, Neil S. AU - Pellicciotti, Francesca ID - 12604 IS - 19 JF - Geophysical Research Letters KW - General Earth and Planetary Sciences KW - Geophysics SN - 0094-8276 TI - Surface pond energy absorption across four Himalayan Glaciers accounts for 1/8 of total catchment ice loss VL - 45 ER - TY - JOUR AB - Supraglacial ice cliffs exist on debris-covered glaciers worldwide, but despite their importance as melt hot spots, their life cycle is little understood. Early field observations had advanced a hypothesis of survival of north-facing and disappearance of south-facing cliffs, which is central for predicting the contribution of cliffs to total glacier mass losses. Their role as windows of energy transfer suggests they may explain the anomalously high mass losses of debris-covered glaciers in High Mountain Asia (HMA) despite the insulating debris, currently at the center of a debated controversy. We use a 3D model of cliff evolution coupled to very high-resolution topographic data to demonstrate that ice cliffs facing south (in the Northern Hemisphere) disappear within a few months due to enhanced solar radiation receipts and that aspect is the key control on cliffs evolution. We reproduce continuous flattening of south-facing cliffs, a result of their vertical gradient of incoming solar radiation and sky view factor. Our results establish that only north-facing cliffs are recurrent features and thus stable contributors to the melting of debris-covered glaciers. Satellite observations and mass balance modeling confirms that few south-facing cliffs of small size exist on the glaciers of Langtang, and their contribution to the glacier volume losses is very small (∼1%). This has major implications for the mass balance of HMA debris-covered glaciers as it provides the basis for new parameterizations of cliff evolution and distribution to constrain volume losses in a region where glaciers are highly relevant as water sources for millions of people. AU - Buri, Pascal AU - Pellicciotti, Francesca ID - 12607 IS - 17 JF - PNAS SN - 0027-8424 TI - Aspect controls the survival of ice cliffs on debris-covered glaciers VL - 115 ER -