Threshold Conjecture Research Articles

Sunflowers: from soil to oil

A sunflower is a collection of sets whose pairwise intersections are identical. In this article, we shall go sunflower-picking. We find sunflowers in several seemingly unrelated fields, before turning to discuss recent progress on the famous sunflower conjecture of Erdős and Rado, made by Alweiss, Lovett, Wu, and Zhang, as well as a related resolution of the threshold vs expectation threshold conjecture of Kahn and Kalai discovered by Park and Pham. We give short proofs for both of these results.

The threshold conjecture for the energy critical hyperbolic Yang–Mills equation

This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical 4+1-dimensional hyperbolic Yang–Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time. Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary topological class at optimal regularity in the third paper, here we perform a blow-up analysis that shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence of a non-trivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.

Transition Threshold for the 3D Couette Flow in Sobolev Space

AbstractIn this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number Re. It was proved that if the initial velocity v0 satisfies for some c0 > 0 independent of Re, then the solution of the 3D Navier‐Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture proposed by Trefethen et al. in 1993. Moreover, we prove that the long‐time dynamics of the solution behaves as: for t ≲ Re, the solution will experience a transient growth from O(Re−1) to O(c0) due to the 3D lift‐up effect; for t ≳ Re1/3, the solution will rapidly converge to a streak solution due to the mixing‐enhanced dissipation effect. © 2020 Wiley Periodicals LLC

The hyperbolic Yang–Mills equation in the caloric gauge : local well-posedness and control of energy-dispersed solutions

This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills equation. We consider topologically trivial solutions in the caloric gauge, which was defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy concentration scale. Moreover, we show that regularity and dispersive behavior of the solution persists as long as energy dispersion is small. We also observe that fixed-time regularity (but not dispersive) properties in the caloric gauge may be transferred to the temporal gauge without any loss, proving as a consequence small data global well-posedness in the temporal gauge. The results in this paper are used in the subsequent papers arXiv:1709.08604, arXiv:1709.08606 to prove the sharp Threshold Theorem in caloric gauge in the trivial topological class, and the dichotomy theorem in arbitrary topological classes.

Transition Threshold for the 2-D Couette Flow in a Finite Channel

In this paper, we study nonlinear stability of the 3D plane Couette flow $(y,0,0)$ at high Reynolds number ${Re}$ in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gg Re^{\frac 13}$ due to the mixing-enhanced dissipation effect. This result confirms the transition threshold conjecture proposed by Trefethen et al.(Science, 261(1993), 578-584). To this end, we develop the resolvent estimate method to establish the space-time estimates for the full linearized Navier-Stokes system around the flow $(V(t,y,z), 0,0)$, where $V(t,y,z)$ is a small perturbation(but independent of $Re$) of the Couette flow $y$.

The Hyperbolic Yang–Mills Equation for Connections in an Arbitrary Topological Class

This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling vs.~Scattering Dichotomy for the energy critical hyperbolic Yang--Mills equation in the $(4+1)$-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang--Mills equation in an arbitrary topological class at an optimal regularity. We generalize the standard notion of a topological class of connections on $\mathbb R^{d}$, defined via a pullback to the one-point compactification $\mathbb S^{d} = \mathbb R^{d} \cup \{\infty\}$, to rough connections with curvature in the critical space $L^{\frac{d}{2}}(\mathbb R^{d})$. Moreover, we provide excision and extension techniques for the Yang--Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang--Mills initial data sets. Combined with the results in the previous paper \cite{OTYM2}, we obtain local well-posedness of the hyperbolic Yang--Mills equation on $\mathbb R^{1+d}$ $(d \geq 4)$ in an arbitrary topological class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case $d = 3$, our techniques provide an alternative proof of the classical finite energy global well-posedness theorem of Klainerman--Machedon \cite{KlMa2}, while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao \cite{TaoYM}. Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence.

A Proof of the Block Model Threshold Conjecture

We study a random graph model called the “stochastic block model” in statistics and the “planted partition model” in theoretical computer science. In its simplest form, this is a random graph with two equal-sized classes of vertices, with a within-class edge probability of q and a between-class edge probability of q′. A striking conjecture of Decelle, Krzkala, Moore and Zdeborova [9], based on deep, non-rigorous ideas from statistical physics, gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if q=a/n and q′=b/n, s=(a−b)/2 and d=(a+b)/2, then Decelle et al. conjectured that it is possible to efficiently cluster in a way correlated with the true partition if s2>d and impossible if s2 Cdlnd for sufficiently large C. In a previous work, we proved that indeed it is information theoretically impossible to cluster if s2 ≤ d and moreover that it is information theoretically impossible to even estimate the model parameters from the graph when s2 d. A different independent proof of the same result was recently obtained by Massoulie [20].

Quantization of Time-Like Energy for Wave Maps into Spheres

In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru (Commun. Math. Phys. 298(1):139–230, 2010; Commun. Math. Phys. 298(1):231–264, 2010), but more specifically into the unit Euclidean sphere {mathbb{S}^{n-1}subsetmathbb{R}^{n}} with {ngeq2}, and study further the dynamics of the sequence of wave maps that are obtained in Sterbenz and Tataru (Commun. Math. Phys. 298(1):231–264, 2010) at the final rescaling for a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space. Similar and stronger results have been recently obtained in the equivariant setting by several authors (Côte, Commun. Pure Appl. Math. 68(11):1946–2004, 2015; Côte, Commun. Pure Appl. Math. 69(4):609–612, 2016; Côte, Am. J. Math. 137(1):139–207, 2015; Côte et al., Am. J. Math. 137(1):209–250, 2015; Krieger, Commun. Math. Phys. 250(3):507–580, 2004), where better control on the dispersive term concentrating on the null boundary of the cone is provided, and in some cases the asymptotic decomposition is shown to hold for all time. Here, however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Rivière (Ann. Math. 149(2):785–829, 1999; Duke Math. J. 111:177–193, 2002), which we make work here in the hyperbolic context of Sterbenz and Tataru (Commun. Math. Phys. 298(1), 231–264, 2010).

A Refined Threshold Theorem for (1 + 2)-Dimensional Wave Maps into Surfaces

The recently established threshold theorem for energy critical wave maps states that wave maps with energy less than that of the ground state (i.e., a minimal energy nontrivial harmonic map) are globally regular and scatter on $${\mathbb{R}^{1+2}}$$ . In this note we give a refinement of this theorem when the target is a closed orientable surface, by taking into account an additional invariant of the problem, namely the topological degree. We show that the sharp energy threshold for global regularity and scattering is in fact twice the energy of the ground state for wave maps with degree zero, whereas wave maps with nonzero degree necessarily have at least the energy of the ground state. We also give a discussion on the formulation of a refined threshold conjecture for the energy critical SU(2) Yang–Mills equation on $${\mathbb{R}^{1+4}}$$ .

The growth threshold conjecture: a theoretical framework for understanding T-cell tolerance.

Adaptive immune responses depend on the capacity of T cells to target specific antigens. As similar antigens can be expressed by pathogens and host cells, the question naturally arises of how can T cells discriminate friends from foes. In this work, we suggest that T cells tolerate cells whose proliferation rates remain below a permitted threshold. Our proposal relies on well-established facts about T-cell dynamics during acute infections: T-cell populations are elastic (they expand and contract) and they display inertia (contraction is delayed relative to antigen removal). By modelling inertia and elasticity, we show that tolerance to slow-growing populations can emerge as a population-scale feature of T cells. This result suggests a theoretical framework to understand immune tolerance that goes beyond the self versus non-self dichotomy. It also accounts for currently unexplained observations, such as the paradoxical tolerance to slow-growing pathogens or the presence of self-reactive T cells in the organism.

A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice

In 1982, Tsallis derived a formula which proposed an exact value of 0.522372078... for the bond percolation threshold of the kagome lattice. We use the substitution method, which is based on stochastic ordering, to compare the probability distribution of connections in the homogeneous bond percolation model on the kagome lattice to those of an exactly-solved inhomogeneous bond percolation model on the martini lattice. The bounds obtained are $0.522394<p_c(\mbox{kagome}) < 0.526750$, where the lower bound shows that the value conjectured by Tsallis is incorrect.

Thresholds and Expectation-Thresholds of Monotone Properties with Small Minterms

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $\mu_p$ on the power set of $N$, where $\mu_p(\cal{A}) := Pr[N_p \in \cal{A}]$ for $\cal{A} \subseteq 2^N$.In this paper we study monotone (upward-closed) families $\cal{A}$ for which all minimal sets in $cal{A}$ have size at most $k$, for some positive integer $k$. We prove that for such a family $\mu_p(\cal{A}) / p^k $ is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables one to identify in $\cal{A}$ either a substantial subfamily $\cal{A}_0$ for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than $k$.Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using linear programming duality. This is related to the threshold conjecture of Kahn and Kalai.

Characterization of large energy solutions of the equivariant wave map problem: I

We consider $1$-equivariant wave maps from ${\Bbb R}^{1+2}\to{\Bbb S}^2$. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, $Q$, given by stereographic projection. We deduce this result via the concentration compactness/rigidity method developed by the second author and Merle. In particular, we establish a classification of equivariant wave maps with trajectories that are pre-compact in the energy space up to the scaling symmetry of the equation. Indeed, a wave map of this type can only be either $0$ or $Q$ up to a rescaling. This gives a proof in the equivariant case of a refined version of the {\it threshold conjecture} adapted to the degree zero theory where the true threshold is $2{\mathcal E}(Q)$, not ${\mathcal E}(Q)$. The aforementioned global existence and scattering statement can also be deduced by considering the work of Sterbenz and Tataru in the equivariant setting. For wave maps of topological degree one, we establish a classification of solutions blowing up in finite time with energies less than three times the energy of $Q$. Under this restriction on the energy, we show that a blow-up solution of degree one is essentially the sum of a rescaled $Q$ plus a remainder term of topological degree zero of energy less than twice the energy of $Q$. This result reveals the universal character of the known blow-up constructions for degree one, $1$-equivariant wave maps of Krieger, the fourth author, and Tataru as well as Rapha\"{e}l and Rodnianski.

Random MAX SAT, random MAX CUT, and their phase transitions

AbstractWith random inputs, certain decision problems undergo a “phase transition.” We prove similar behavior in an optimization context. Given a conjunctive normal form (CNF) formula F on n variables and with m k‐variable clauses, denote by max F the maximum number of clauses satisfiable by a single assignment of the variables. (Thus the decision problem k‐SAT is to determine if max F is equal to m.) With the formula F chosen at random, the expectation of max F is trivially bounded by (3/4)m ⩽ 𝔼 max F ⩽ m. We prove that for random formulas with m = ⌊cn⌋ clauses: for constants c < 1, 𝔼 max F is ⌊cn⌋ − Θ(1/n); for large c, it approaches and in the “window” c = 1 + Θ(n−1/3), it is cn − Θ(1). Our full results are more detailed, but this already shows that the optimization problem MAX 2‐SAT undergoes a phase transition just as the 2‐SAT decision problem does, and at the same critical value c = 1. Most of our results are established without reference to the analogous propositions for decision 2‐SAT, and can be used to reproduce them. We consider “online” versions of MAX 2‐SAT, and show that for one version the obvious greedy algorithm is optimal; all other natural questions remain open. We can extend only our simplest MAX 2‐SAT results to MAX k‐SAT, but we conjecture a “MAX k‐SAT limiting function conjecture” analogous to the folklore “satisfiability threshold conjecture,” but open even for k = 2. Neither conjecture immediately implies the other, but it is natural to further conjecture a connection between them. We also prove analogous results for random MAX CUT. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

Bounding the unsatisfiability threshold of random 3-SAT

The satisfiability threshold conjecture states that fur a randomly generated formula of m clauses of exactly k literals over n variables, the probability that it is satisfiable, as n tends to infinity, changes abruptly from I to 0, as the ratio I = m/n is

General practitioners' referral thresholds and choices of referral destination: an experimental study.

General practitioners (GPs) exert a major impact on NHS resource use, both as providers of primary care and as referrers to secondary care. Referral rates are subject to wide variations, leading to the conjecture that certain GPs may have different 'referral thresholds' from those of others. In this paper, the authors describe an experiment designed both to test the referral threshold hypothesis and to illuminate the GP's decision process with respect to choice over referral destination. Nottinghamshire GPs were provided with hypothetical case histories and a list of possible referral destinations, specifying a range of consultants, their specialist interests, plus the expected waiting times and costs for both out-patient investigation and in-patient treatment. For each case, respondents were requested to indicate whether or not they would refer the patient, and to whom. Respondents were also asked to indicate the extent to which their choices of consultants generally were governed by the specialist interest, the waiting time and the cost information. The responses of the sample support the referral threshold conjecture, with specialist interests and waiting time appearing to be far more important than cost in influencing choice of referral destination. The possibilities of influencing GPs' referral behaviour are discussed, in the light of recent initiatives with respect to prescribing.

Tail bounds for occupancy and the satisfiability threshold conjecture

AbstractThe classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and probabilistic analysis. Our motivating application is the following well‐known conjecture on threshold phenomenon for the satisfiability problem. Consider random 3‐SAT formulas with cn clauses over n variables, where each clause is chosen uniformly and independently from the space of all clauses of size 3. It has been conjectured that there is a sharp threshold for satisfiability at c* ≈︁ 4.2. We provide a strong upper bound on the value of c*, showing that for c > 4.758 a random 3‐SAT formula is unsatisfiable with high probability. This result is based on a structural property, possibly of independent interest, whose proof needs several applications of the occupancy tail bounds.