arXiv subjects

Jeong Han Kim

Publications and source records attributed to Jeong Han Kim.

52 records · Page 3Linked to original sources

arXiv2015

Search Strategies for TeV Scale Fermionic Top Partners with Charge 2/3

Searches for fermionic top partners at the TeV scale will bring forward a new final state kinematic regime and event topologies, where Run I search strategies will inevitably fail. We propose concrete search strategies for singly produced charge 2/3 fermionic top partners ($T'$) adequate for LHC Run II. Our analysis spans over all of the $T'$ decay modes ($i.e.$ $tZ$, $th$ and $Wb$) where we present detailed discussion of the search performances, signal efficiencies and backgrounds rates. Our LHC Run II search proposals utilize signatures with large missing energy and leptons, as well as jet substructure observables for tagging of boosted heavy SM states, customized $b$-tagging tactics and forward jet tagging. We analyze the prospects for discovery and exclusion of $T'$ models within the framework of partially composite quarks at the LHC Run II. Our results show that the LHC Run II has good prospects for observing $T'$ models which predict single production cross section of $σ_{T'} \sim 70 -140 \, (30 -65)$ fb for $M_{T'} = 1\, (1.5)$~TeV respectively with 100 fb$^{-1}$ of integrated luminosity, depending on the branching ratios of the $T'$. Similarly, we find that cross sections of $σ_{T'} \sim 27 -60\, (13 - 24)$ fb for $M_{T'} = 1\, (1.5)$ TeV respectively can be excluded with the same amount of data. Our results are minimally model dependent and can be applied to most $T'$ models where $Γ_{T'} \ll M_{T'}$.

hep-ph↗

arXiv2015

Boosted Event Topologies from TeV Scale Light Quark Composite Partners

We propose a new search strategy for quark partners which decay into a boosted Higgs and a light quark. As an example, we consider phenomenologically viable right handed up-type quark partners of mass $\sim 1$ TeV in composite pseudo-Nambu-Goldstone-boson Higgs models within the context of flavorful naturalness. Our results show that $S/B > 1$ and signal significance of $\sim 7σ$ is achievable at $\sqrt{s} = 14$ TeV LHC with 35 $fb^{-1}$ of integrated luminosity, sufficient to claim discovery of a new particle. A combination of a multi-dimensional boosted Higgs tagging technique, kinematics of pair produced heavy objects and $b$-tagging serves to efficiently diminish the large QCD backgrounds while maintaining adequate levels of signal efficiency. We present the analysis in the context of effective field theory, such that our results can be applied to any future search for pair produced vector-like quarks with decay modes to Higgs and a light jet.

hep-ph↗

arXiv2014

Two Approaches to Sidorenko's Conjecture

Sidorenko's conjecture states that for every bipartite graph $H$ on $\{1,\cdots,k\}$, $\int \prod_{(i,j)\in E(H)} h(x_i, y_j) dμ^{|V(H)|} \ge \left( \int h(x,y) \,dμ^2 \right)^{|E(H)|}$ holds, where $μ$ is the Lebesgue measure on $[0,1]$ and $h$ is a bounded, non-negative, symmetric, measurable function on $[0,1]^2$. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph $H$ to a graph $G$ is asymptotically at least the expected number of homomorphisms from $H$ to the Erdős-Rényi random graph with the same expected edge density as $G$. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph $H$ with bipartition $A \cup B$ is tree-arrangeable if neighborhoods of vertices in $A$ have a certain tree-like structure. We show that Sidorenko's conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's conjecture holds if there are two vertices $a_1, a_2$ in $A$ such that each vertex $a \in A$ satisfies $N(a) \subseteq N(a_1)$ or $N(a) \subseteq N(a_2)$, and also implies a recent result of Conlon, Fox, and Sudakov \cite{CoFoSu}. Second, if $T$ is a tree and $H$ is a bipartite graph satisfying Sidorenko's conjecture, then it is shown that the Cartesian product $T \Box H$ of $T$ and $H$ also satisfies Sidorenko's conjecture. This result implies that, for all $d \ge 2$, the $d$-dimensional grid with arbitrary side lengths satisfies Sidorenko's conjecture.

math.CO↗

arXiv2014

On Coupon Colorings of Graphs

Let $G$ be a graph with no isolated vertices. A {\em $k$-coupon coloring} of $G$ is an assignment of colors from $[k] := \{1,2,\dots,k\}$ to the vertices of $G$ such that the neighborhood of every vertex of $G$ contains vertices of all colors from $[k]$. The maximum $k$ for which a $k$-coupon coloring exists is called the {\em coupon coloring number} of $G$, and is denoted $χ_{c}(G)$. In this paper, we prove that every $d$-regular graph $G$ has $χ_{c}(G) \geq (1 - o(1))d/\log d$ as $d \rightarrow \infty$, and the proportion of $d$-regular graphs $G$ for which $χ_c(G) \leq (1 + o(1))d/\log d$ tends to $1$ as $|V(G)| \rightarrow \infty$.

math.CO↗

arXiv2014

Constraints on composite quark partners from Higgs searches

In composite Higgs models, the generation of quark masses requires the standard model-like quarks to be partially or fully composite states which are accompanied by composite quark partners. The composite quark partners decay into a standard model-like quark and an electroweak gauge boson or Higgs boson, which can be searched for at the LHC. In this article, we study the phenomenological implications of composite quarks in the minimal composite Higgs model based on the coset $SO(5)/SO(4)$. We focus on light quark partners which are embedded in the $SO(4)$ singlet representation. In this case, a dominant decay mode of the partner quark is into a Higgs boson and a jet, for which no experimental bounds have been established so far. The presence of $SO(4)$ singlet partners leads to an enhancement of the di-Higgs production cross section at the LHC. This will be an interesting experimental signature in the near future, but, unfortunately, there are no direct bounds available yet from the experimental analyses. However, we find that the currently available standard model-like Higgs searches can be used in order to obtain the first constraints on partially and fully composite quark models with light quark partners in the $SO(4)$ singlet. We obtain a flavor- and composition parameter independent bound on the quark partner mass of $M_{U_h} > 310 \mbox{ GeV}$ for partially composite quark models and $M_{U_h} > 212 \mbox{ GeV}$ for fully composite quark models.

hep-ph↗

arXiv2013

Universality of random graphs for graphs of maximum degree two

For a family $\mathcal{F}$ of graphs, a graph $G$ is called \emph{$\mathcal{F}$-universal} if $G$ contains every graph in $\mathcal{F}$ as a subgraph. Let $\mathcal{F}_n(d)$ be the family of all graphs on $n$ vertices with maximum degree at most $d$. Dellamonica, Kohayakawa, Rödl and Ruciński showed that, for $d\geq 3$, the random graph $G(n,p)$ is $\mathcal{F}_n(d)$-universal with high probability provided $p\geq C\big(\frac{\log n}{n}\big)^{1/d}$ for a sufficiently large constant $C=C(d)$. In this paper we prove the missing part of the result, that is, the random graph $G(n,p)$ is $\mathcal{F}_n(2)$-universal with high probability provided $p\geq C\big(\frac{\log n}{n}\big)^{1/2}$ for a sufficiently large constant $C$.

math.CO↗

arXiv2012

A Sequential Algorithm for Generating Random Graphs

We present a nearly-linear time algorithm for counting and randomly generating simple graphs with a given degree sequence in a certain range. For degree sequence $(d_i)_{i=1}^n$ with maximum degree $d_{\max}=O(m^{1/4-τ})$, our algorithm generates almost uniform random graphs with that degree sequence in time $O(m\,d_{\max})$ where $m=\f{1}{2}\sum_id_i$ is the number of edges in the graph and $τ$ is any positive constant. The fastest known algorithm for uniform generation of these graphs McKay Wormald (1990) has a running time of $O(m^2d_{\max}^2)$. Our method also gives an independent proof of McKay's estimate McKay (1985) for the number of such graphs. We also use sequential importance sampling to derive fully Polynomial-time Randomized Approximation Schemes (FPRAS) for counting and uniformly generating random graphs for the same range of $d_{\max}=O(m^{1/4-τ})$. Moreover, we show that for $d = O(n^{1/2-τ})$, our algorithm can generate an asymptotically uniform $d$-regular graph. Our results improve the previous bound of $d = O(n^{1/3-τ})$ due to Kim and Vu (2004) for regular graphs.

cs.CC↗

arXiv2012

Finding Weighted Graphs by Combinatorial Search

We consider the problem of finding edges of a hidden weighted graph using a certain type of queries. Let $G$ be a weighted graph with $n$ vertices. In the most general setting, the $n$ vertices are known and no other information about $G$ is given. The problem is finding all edges of $G$ and their weights using additive queries, where, for an additive query, one chooses a set of vertices and asks the sum of the weights of edges with both ends in the set. This model has been extensively used in bioinformatics including genom sequencing. Extending recent results of Bshouty and Mazzawi, and Choi and Kim, we present a polynomial time randomized algorithm to find the hidden weighted graph $G$ when the number of edges in $G$ is known to be at most $m\geq 2$ and the weight $w(e)$ of each edge $e$ satisfies $\ga \leq |w(e)|\leq \gb$ for fixed constants $\ga, \gb>0$. The query complexity of the algorithm is $O(\frac{m \log n}{\log m})$, which is optimal up to a constant factor.

math.CO↗

arXiv2010

A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm

We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group $G$ and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in $Θ(\sqrt{|G|})$ steps. Moreover, for the parallelized distinguished points algorithm on $J$ processors we find that $Θ(\sqrt{|G|}/J)$ steps suffices. These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from $G$.

math.PR↗

arXiv2010

Diameters in supercritical random graphs via first passage percolation

We study the diameter of $C_1$, the largest component of the Erdős-Rényi random graph $G(n,p)$ in the emerging supercritical phase, i.e., for $p = \frac{1+ε}n$ where $ε^3 n \to \infty$ and $ε=o(1)$. This parameter was extensively studied for fixed $ε> 0$, yet results for $ε=o(1)$ outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter, however these did not cover the entire supercritical regime (namely, when $ε^3 n\to\infty$ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of $1000/7$. We show that throughout the emerging supercritical phase, i.e. for any $ε=o(1)$ with $ε^3 n \to \infty$, the diameter of $C_1$ is with high probability asymptotic to $D(ε,n)=(3/ε)\log(ε^3 n)$. This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of $C_1$ is w.h.p. asymptotic to $(2/3)D(ε,n)$, and the maximal distance in $C_1$ between any pair of kernel vertices is w.h.p. asymptotic to $(5/9)D(ε,n)$.

math.CO↗

arXiv2009

Anatomy of a young giant component in the random graph

We provide a complete description of the giant component of the Erdős-Rényi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+ε)/n$ where $ε^3 n \to \infty$ and $ε=o(1)$. Our description is particularly simple for $ε= o(n^{-1/4})$, where the giant component $C_1$ is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for $C_1$). Let $Z$ be normal with mean $\frac23 ε^3 n$ and variance $ε^3 n$, and let $K$ be a random 3-regular graph on $2\lfloor Z\rfloor$ vertices. Replace each edge of $K$ by a path, where the path lengths are i.i.d. geometric with mean $1/ε$. Finally, attach an independent Poisson($1-ε$)-Galton-Watson tree to each vertex. A similar picture is obtained for larger $ε=o(1)$, in which case the random 3-regular graph is replaced by a random graph with $N_k$ vertices of degree $k$ for $k\geq 3$, where $N_k$ has mean and variance of order $ε^k n$. This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of $ε$, as well as the mixing time of the random walk on $C_1$.

math.CO↗

arXiv2008

Finding cores of random 2-SAT formulae via Poisson cloning

For the random 2-SAT formula $F(n,p)$, let $F_C (n,p)$ be the formula left after the pure literal algorithm applied to $F(n,p)$ stops. Using the recently developed Poisson cloning model together with the cut-off line algorithm (COLA), we completely analyze the structure of $F_{C} (n,p)$. In particular, it is shown that, for $\gl:= p(2n-1) = 1+\gs $ with $\gs\gg n^{-1/3}$, the core of $F(n,p)$ has $\thl^2 n +O((\thl n)^{1/2})$ variables and $\thl^2 \gl n+O((\thl n))^{1/2}$ clauses, with high probability, where $\thl$ is the larger solution of the equation $þ- (1-e^{-\thl \gl})=0$. We also estimate the probability of $F(n,p)$ being satisfiable to obtain $$ \pr[ F_2(n, \sfrac{\gl}{2n-1}) is satisfiable ] = \caseth{1-\frac{1+o(1)}{16\gs^3 n}}{if $\gl= 1-\gs$ with $\gs\gg n^{-1/3}$}{}{}{e^{-Θ(\gs^3n)}}{if $\gl=1+\gs$ with $\gs\gg n^{-1/3}$,} $$ where $o(1)$ goes to 0 as $\gs$ goes to 0. This improves the bounds of Bollobás et al. \cite{BBCKW}.

math.CO↗

arXiv2008

Poisson Cloning Model for Random Graphs

In the random graph $G(n,p)$ with $pn$ bounded, the degrees of the vertices are almost i.i.d Poisson random variables with mean $\gl:= p(n-1)$. Motivated by this fact, we introduce the Poisson cloning model $G_{PC} (n,p)$ for random graphs in which the degrees are i.i.d Poisson random variables with mean $\gl$. Then, we first establish a theorem that shows the new model is equivalent to the classical model $G(n,p)$ in an asymptotic sense. Next, we introduce a useful algorithm, called the cut-off line algorithm, to generate the random graph $G_{PC} (n,p)$. The Poisson cloning model $G_{PC}(n,p)$ equipped with the cut-off line algorithm enables us to very precisely analyze the sizes of the largest component and the $t$-core of $G(n,p)$. This new approach to the problems yields not only elegant proofs but also improved bounds that are essentially best possible. We also consider the Poisson cloning models for random hypergraphs and random $k$-SAT problems. Then, the $t$-core problem for random hypergraphs and the pure literal algorithm for random $k$-SAT problems are analyzed.

math.CO↗

arXiv2007

Near Optimal Bounds for Collision in Pollard Rho for Discrete Log

We analyze a fairly standard idealization of Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G. It is found that, with high probability, a collision occurs in $O(\sqrt{|G|\log |G| \log \log |G|})$ steps, not far from the widely conjectured value of $Θ(\sqrt{|G|})$. This improves upon a recent result of Miller--Venkatesan which showed an upper bound of $O(\sqrt{|G|}\log^3 |G|)$. Our proof is based on analyzing an appropriate nonreversible, non-lazy random walk on a discrete cycle of (odd) length |G|, and showing that the mixing time of the corresponding walk is $O(\log |G| \log \log |G|)$.

math.NT↗

arXiv2004

How Complex are Random Graphs in First Order Logic?

It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number $(G) of nested quantifiers in a such formula can serve as a measure for the ``first order complexity'' of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is Θ(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log^* n, the inverse of the tower-function. The general picture, however, is still a mystery.

math.CO↗

arXiv2001

The Scaling Window of the 2-SAT Transition

We consider the random 2-satisfiability problem, in which each instance is a formula that is the conjunction of m clauses of the form (x or y), chosen uniformly at random from among all 2-clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n --> alpha, the problem is known to have a phase transition at alpha_c = 1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite-size scaling about this transition, namely the scaling of the maximal window W(n,delta) = (alpha_-(n,delta),alpha_+(n,delta)) such that the probability of satisfiability is greater than 1-delta for alpha < alpha_- and is less than delta for alpha > alpha_+. We show that W(n,delta)=(1-Theta(n^{-1/3}),1+Theta(n^{-1/3})), where the constants implicit in Theta depend on delta. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+epsilon)n, where epsilon may depend on n as long as |epsilon| is sufficiently small and |epsilon|*n^(1/3) is sufficiently large, we show that the probability of satisfiability decays like exp(-Theta(n*epsilon^3)) above the window, and goes to one like 1-Theta(1/(n*|epsilon|^3)) below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2-SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2-SAT are identical to those of the random graph.

math.CO↗