Mostrando recursos 1 - 20 de 2.414

  1. Scaling limits for some random trees constructed inhomogeneously

    Ross, Nathan; Wen, Yuting
    We define some new sequences of recursively constructed random combinatorial trees, and show that, after properly rescaling graph distance and equipping the trees with the uniform measure on vertices, each sequence converges almost surely to a real tree in the Gromov-Hausdorff-Prokhorov sense. The limiting real trees are constructed via line-breaking the half real-line with a Poisson process having rate $(\ell +1)t^\ell dt$, for each positive integer $\ell $, and the growth of the combinatorial trees may be viewed as an inhomogeneous generalization of Rémy’s algorithm.

  2. Stein approximation for functionals of independent random sequences

    Privault, Nicolas; Serafin, Grzegorz
    We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and functionals of both continuous and discrete independent random variables. For random variables admitting a continuous density, it recovers classical distance bounds based on absolute third moments, with better and explicit constants. We also apply this method to multiple stochastic integrals that can be used to represent $U$-statistics, and include linear and quadratic functionals as particular cases.

  3. On the strange domain of attraction to generalized Dickman distributions for sums of independent random variables

    Pinsky, Ross G
    Let $\{B_k\}_{k=1}^\infty , \{X_k\}_{k=1}^\infty $ all be independent random variables. Assume that $\{B_k\}_{k=1}^\infty $ are $\{0,1\}$-valued Bernoulli random variables satisfying $B_k\stackrel{\text {dist}} {=}\text{Ber} (p_k)$, with $\sum _{k=1}^\infty p_k=\infty $, and assume that $\{X_k\}_{k=1}^\infty $ satisfy $X_k>0$ and $\mu _k\equiv EX_k<\infty $. Let $M_n=\sum _{k=1}^np_k\mu _k$, assume that $M_n\to \infty $ and define the normalized sum of independent random variables $W_n=\frac 1{M_n}\sum _{k=1}^nB_kX_k$. We give a general condition under which $W_n\stackrel{\text {dist}} {\to }c$, for some $c\in [0,1]$, and a general condition under which $W_n$ converges weakly to a distribution from a family of distributions that includes the generalized Dickman distributions...

  4. Fluctuations of the empirical measure of freezing Markov chains

    Bouguet, Florian; Cloez, Bertrand
    In this work, we consider a finite-state inhomogeneous-time Markov chain whose probabilities of transition from one state to another tend to decrease over time. This can be seen as a cooling of the dynamics of an underlying Markov chain. We are interested in the long time behavior of the empirical measure of this freezing Markov chain. Some recent papers provide almost sure convergence and convergence in distribution in the case of the freezing speed $n^{-\theta }$, with different limits depending on $\theta <1,\theta =1$ or $\theta >1$. Using stochastic approximation techniques, we generalize these results for any freezing speed, and...

  5. On the critical probability in percolation

    Janson, Svante; Warnke, Lutz
    For percolation on finite transitive graphs, Nachmias and Peres suggested a characterization of the critical probability based on the logarithmic derivative of the susceptibility. As a first test-case, we study their suggestion for the Erdős–Rényi random graph $G_{n,p}$, and confirm that the logarithmic derivative has the desired properties: (i) its maximizer lies inside the critical window $p=1/n+\Theta (n^{-4/3})$, and (ii) the inverse of its maximum value coincides with the $\Theta (n^{-4/3})$–width of the critical window. We also prove that the maximizer is not located at $p=1/n$ or $p=1/(n-1)$, refuting a speculation of Peres.

  6. On percolation critical probabilities and unimodular random graphs

    Beringer, Dorottya; Pete, Gábor; Timár, Ádám
    We investigate generalizations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\tilde{p} _c$ defined by Duminil-Copin and Tassion [11] to bounded degree unimodular random graphs. We further examine Schramm’s conjecture in the case of unimodular random graphs: does ${p_c}(G_n)$ converge to ${p_c}(G)$ if $G_n\to G$ in the local weak sense? Among our results are the following: [start-list] *${p_c}={\tilde{p} _c}$ holds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and ${p_T}<{p_c}$; i.e., the classical sharpness of phase transition does not hold. *We give conditions which imply $\lim{p_c} (G_n)= {p_c}(\lim G_n)$. *There...

  7. On percolation critical probabilities and unimodular random graphs

    Beringer, Dorottya; Pete, Gábor; Timár, Ádám
    We investigate generalizations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\tilde{p} _c$ defined by Duminil-Copin and Tassion [11] to bounded degree unimodular random graphs. We further examine Schramm’s conjecture in the case of unimodular random graphs: does ${p_c}(G_n)$ converge to ${p_c}(G)$ if $G_n\to G$ in the local weak sense? Among our results are the following: [start-list] *${p_c}={\tilde{p} _c}$ holds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and ${p_T}<{p_c}$; i.e., the classical sharpness of phase transition does not hold. *We give conditions which imply $\lim{p_c} (G_n)= {p_c}(\lim G_n)$. *There...

  8. Percolation and convergence properties of graphs related to minimal spanning forests

    Hirsch, Christian; Brereton, Tim; Schmidt, Volker
    Lyons, Peres and Schramm have shown that minimal spanning forests on randomly weighted lattices exhibit a critical geometry in the sense that adding or deleting only a small number of edges results in a radical change of percolation properties. We show that these results can be extended to a Euclidean setting by considering families of stationary super- and subgraphs that approximate the Euclidean minimal spanning forest arbitrarily closely, but whose percolation properties differ decisively from those of the minimal spanning forest. Since these families can be seen as generalizations of the relative neighborhood graph and the nearest-neighbor graph, respectively, our...

  9. Percolation and convergence properties of graphs related to minimal spanning forests

    Hirsch, Christian; Brereton, Tim; Schmidt, Volker
    Lyons, Peres and Schramm have shown that minimal spanning forests on randomly weighted lattices exhibit a critical geometry in the sense that adding or deleting only a small number of edges results in a radical change of percolation properties. We show that these results can be extended to a Euclidean setting by considering families of stationary super- and subgraphs that approximate the Euclidean minimal spanning forest arbitrarily closely, but whose percolation properties differ decisively from those of the minimal spanning forest. Since these families can be seen as generalizations of the relative neighborhood graph and the nearest-neighbor graph, respectively, our...

  10. Stochastic complex Ginzburg-Landau equation with space-time white noise

    Hoshino, Masato; Inahama, Yuzuru; Naganuma, Nobuaki
    We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be understood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.

  11. Stochastic complex Ginzburg-Landau equation with space-time white noise

    Hoshino, Masato; Inahama, Yuzuru; Naganuma, Nobuaki
    We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be understood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.

  12. Stochastic complex Ginzburg-Landau equation with space-time white noise

    Hoshino, Masato; Inahama, Yuzuru; Naganuma, Nobuaki
    We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be understood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.

  13. Branching Brownian motion, mean curvature flow and the motion of hybrid zones

    Etheridge, Alison; Freeman, Nic; Penington, Sarah
    We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $\Lambda $-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.

  14. Branching Brownian motion, mean curvature flow and the motion of hybrid zones

    Etheridge, Alison; Freeman, Nic; Penington, Sarah
    We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $\Lambda $-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.

  15. Branching Brownian motion, mean curvature flow and the motion of hybrid zones

    Etheridge, Alison; Freeman, Nic; Penington, Sarah
    We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $\Lambda $-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.

  16. Branching Brownian motion, mean curvature flow and the motion of hybrid zones

    Etheridge, Alison; Freeman, Nic; Penington, Sarah
    We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $\Lambda $-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.

  17. Extreme statistics of non-intersecting Brownian paths

    Nguyen, Gia Bao; Remenik, Daniel
    We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [28] for the joint distribution of $\mathcal{M} =\max _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$ and $\mathcal{T} =\operatorname{argmax} _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$, where $\mathcal{A} _2$ is the Airy$_2$ process, and we use them to show...

  18. Extreme statistics of non-intersecting Brownian paths

    Nguyen, Gia Bao; Remenik, Daniel
    We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [28] for the joint distribution of $\mathcal{M} =\max _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$ and $\mathcal{T} =\operatorname{argmax} _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$, where $\mathcal{A} _2$ is the Airy$_2$ process, and we use them to show...

  19. Extreme statistics of non-intersecting Brownian paths

    Nguyen, Gia Bao; Remenik, Daniel
    We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [28] for the joint distribution of $\mathcal{M} =\max _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$ and $\mathcal{T} =\operatorname{argmax} _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$, where $\mathcal{A} _2$ is the Airy$_2$ process, and we use them to show...

  20. Extreme statistics of non-intersecting Brownian paths

    Nguyen, Gia Bao; Remenik, Daniel
    We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [28] for the joint distribution of $\mathcal{M} =\max _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$ and $\mathcal{T} =\operatorname{argmax} _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$, where $\mathcal{A} _2$ is the Airy$_2$ process, and we use them to show...

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