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1. CJM Online first

Müllner, Clemens
The Rudin-Shapiro sequence and similar sequences are normal along squares
We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.

Keywords:Rudin-Shapiro sequence, digital sequence, normality, exponential sum
Categories:11A63, 11B85, 11L03, 11N60, 60F05

2. CJM Online first

Mingo, James A.; Popa, Mihai
Freeness and The Partial Transposes of Wishart Random Matrices
We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives a example where the partial transpose produces freeness at the operator level. Finally we investigate the case of real Wishart matrices.

Keywords:free probability, random matrix, partial transpose, quantum information theory
Categories:15B52, 46L54, 60B20

3. CJM Online first

Bernardi, Olivier; Curien, Nicolas; Miermont, Grégory
A Boltzmann approach to percolation on random triangulations
We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length $n$ decays exponentially with $n$ except at a particular value $p_c$ of the percolation parameter $p$ for which the decay is polynomial (of order $n^{-10/3}$). Moreover, the probability that the origin cluster has size $n$ decays exponentially if $p\lt p_c$ and polynomially if $p\geq p_c$. The critical percolation value is $p_c=1/2$ for site percolation, and $p_c=\frac{2\sqrt{3}-1}{11}$ for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds. Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at $p_c$, the percolation clusters conditioned to have size $n$ should converge toward the stable map of parameter $ \frac{7}{6}$ introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

Keywords:random map, stable map, critical percolation, gasket
Categories:60K35, 60D05, 05A16

4. CJM Online first

Barlow, Martin T.; Járai, Antal A.
Geometry of uniform spanning forest components in high dimensions
We study the geometry of the component of the origin in the uniform spanning forest of $\mathbb{Z}^d$ and give bounds on the size of balls in the intrinsic metric.

Keywords:uniform spanning forest, loop-erased random walk
Categories:60D05, 05C05, 31C20

5. CJM 2017 (vol 70 pp. 3)

Benaych-Georges, Florent; Cébron, Guillaume; Rochet, Jean
Fluctuation of matrix entries and application to outliers of elliptic matrices
For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in K}$ which is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type $\operatorname{Tr}(\mathbf{A}_k \mathbf{M})$, where the matrix $\mathbf{M}$ is deterministic (such random variables include for example the normalized matrix entries of the $\mathbf{A}_k$'s). A consequence is the asymptotic independence of the projection of the matrices $\mathbf{A}_k$ onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Form of the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other. These phenomena have already been observed with random matrices from the Single Ring Theorem.

Keywords:random matrix, Gaussian fluctuation, spiked model, elliptic random matrix, Weingarten calculus, Haar measure
Categories:60B20, 15B52, 60F05, 46L54

6. CJM Online first

Handelman, David
Nearly approximate transitivity (AT) for circulant matrices
By previous work of Giordano and the author, ergodic actions of $\mathbf Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.

Keywords:approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrix-valued random walk
Categories:37A05, 06F25, 28D05, 46B40, 60G50

7. CJM 2017 (vol 69 pp. 481)

Cordero-Erausquin, Dario
Transport Inequalities for Log-concave Measures, Quantitative Forms and Applications
We review some simple techniques based on monotone mass transport that allow us to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.

Keywords:log-concave measures, transport inequality, Brascamp-Lieb inequality, quantitative inequalities
Categories:52A40, 60E15, 49Q20

8. CJM 2015 (vol 68 pp. 129)

Shiozawa, Yuichi
Lower Escape Rate of Symmetric Jump-diffusion Processes
We establish an integral test on the lower escape rate of symmetric jump-diffusion processes generated by regular Dirichlet forms. Using this test, we can find the speed of particles escaping to infinity. We apply this test to symmetric jump processes of variable order. We also derive the upper and lower escape rates of time changed processes by using those of underlying processes.

Keywords:lower escape rate, Dirichlet form, Markov process, time change
Categories:60G17, 31C25, 60J25

9. CJM 2014 (vol 66 pp. 1358)

Osėkowski, Adam
Sharp Localized Inequalities for Fourier Multipliers
In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator .

Keywords:Fourier multiplier, martingale, laminate
Categories:42B15, 60G44, 42B20

10. CJM 2013 (vol 66 pp. 1050)

Holmes, Mark; Salisbury, Thomas S.
Random Walks in Degenerate Random Environments
We study the asymptotic behaviour of random walks in i.i.d. random environments on $\mathbb{Z}^d$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience, and in 2-dimensions the existence of a deterministic limiting velocity.

Keywords:random walk, non-elliptic random environment, zero-one law, coupling
Category:60K37

11. CJM 2012 (vol 65 pp. 600)

Kroó, A.; Lubinsky, D. S.
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials
We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain - in particular this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex, under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.

Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions
Categories:42C05, 42C99, 42B05, 60B20

12. CJM 2011 (vol 64 pp. 1201)

Aistleitner, Christoph; Elsholtz, Christian
The Central Limit Theorem for Subsequences in Probabilistic Number Theory
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying \begin{equation} \tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty. \end{equation} If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of \begin{equation} \tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}} \end{equation} converges to a Gaussian distribution. In the case $$ 1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty $$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$ \frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}} $$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem
Categories:60F05, 42A55, 11D04, 05C55, 11K06

13. CJM 2011 (vol 64 pp. 1075)

Raja, Chandiraraj Robinson Edward
A Stochastic Difference Equation with Stationary Noise on Groups
We consider the stochastic difference equation $$\eta _k = \xi _k \phi (\eta _{k-1}), \quad k \in \mathbb Z $$ on a locally compact group $G$ where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$-valued random variables and $\eta _k$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on one-dimensional torus. We consider the case when $\xi _k$ have a common law $\mu$ and prove that if $G$ is a distal group and $\phi$ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu$ is supported on $Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

Keywords:dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powers
Categories:60B15, 60G20

14. CJM 2011 (vol 64 pp. 961)

Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim
Densities of Short Uniform Random Walks
We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

Keywords:random walks, hypergeometric functions, Mahler measure
Categories:60G50, 33C20, 34M25, 44A10

15. CJM 2011 (vol 64 pp. 805)

Chapon, François; Defosseux, Manon
Quantum Random Walks and Minors of Hermitian Brownian Motion
Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.

Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process
Categories:46L53, 60B20, 14L24

16. CJM 2011 (vol 64 pp. 869)

Hu, Ze-Chun; Sun, Wei
Balayage of Semi-Dirichlet Forms
In this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayaged measures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting.

Keywords:balayage, semi-Dirichlet form, potential theory
Categories:31C25, 60J45

17. CJM 2010 (vol 63 pp. 153)

Hambly, B. M.
Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet
We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function, and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$ and $\phi$ a periodic function. We also discuss similar results for the heat content of affine nested fractals.

Categories:35K05, 28A80, 35B40, 60J65

18. CJM 2010 (vol 63 pp. 222)

Wang, Jiun-Chau
Limit Theorems for Additive Conditionally Free Convolution
In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution.

Keywords:additive c-free convolution, limit theorems, infinitesimal arrays
Categories:46L53, 60F05

19. CJM 2010 (vol 63 pp. 104)

Feng, Shui; Schmuland, Byron; Vaillancourt, Jean; Zhou, Xiaowen
Reversibility of Interacting Fleming-Viot Processes with Mutation, Selection, and Recombination
Reversibility of the Fleming--Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming--Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.

Categories:60J60, 60J70

20. CJM 2009 (vol 61 pp. 1279)

Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval
Tail Bounds for the Stable Marriage of Poisson and Lebesgue
Let $\Xi$ be a discrete set in $\rd$. Call the elements of $\Xi$ {\em centers}. The well-known Voronoi tessellation partitions $\rd$ into polyhedral regions (of varying volumes) by allocating each site of $\rd$ to the closest center. Here we study allocations of $\rd$ to $\Xi$ in which each center attempts to claim a region of equal volume $\alpha$. We focus on the case where $\Xi$ arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is {\em stable} in the sense of the Gale--Shapley marriage problem. We study the distance $X$ from a typical site to its allocated center in the stable allocation. The model exhibits a phase transition in the appetite $\alpha$. In the critical case $\alpha=1$ we prove a power law upper bound on $X$ in dimension $d=1$. (Power law lower bounds were proved earlier for all $d$). In the non-critical cases $\alpha<1$ and $\alpha>1$ we prove exponential upper bounds on $X$.

Keywords:stable marriage, point process, phase transition
Category:60D05

21. CJM 2009 (vol 61 pp. 534)

Chen, Chuan-Zhong; Sun, Wei
Girsanov Transformations for Non-Symmetric Diffusions
Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form $(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(E;m)$. For $u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process $Y$ of $X$. First, let $\widehat{X}$ be the dual process of $X$ and $\widehat{Y}$ the Girsanov transformed process of $\widehat{X}$. We give a necessary and sufficient condition for $(Y,\widehat{Y})$ to be in duality with respect to the measure $e^{2u}m$. We also construct a counterexample, which shows that this condition may not be satisfied and hence $(Y,\widehat{Y})$ may not be dual processes. Then we present a sufficient condition under which $Y$ is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Keywords:Diffusion, non-symmetric Dirichlet form, Girsanov transformation, $h$-transformation, perturbation of Dirichlet form, generalized Feynman-Kac semigroup
Categories:60J45, 31C25, 60J57

22. CJM 2008 (vol 60 pp. 822)

Kuwae, Kazuhiro
Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms
Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Keywords:positivity preserving form, semi-Dirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity condition
Categories:31C25, 35B50, 60J45, 35J, 53C, 58

23. CJM 2008 (vol 60 pp. 334)

Curry, Eva
Low-Pass Filters and Scaling Functions for Multivariable Wavelets
We show that a characterization of scaling functions for multiresolution analyses given by Hern\'{a}ndez and Weiss and that a characterization of low-pass filters given by Gundy both hold for multivariable multiresolution analyses.

Keywords:multivariable multiresolution analysis, low-pass filter, scaling function
Categories:42C40, 60G35

24. CJM 2008 (vol 60 pp. 313)

Choi, Yong-Kab; o, Miklós Csörg\H
Asymptotic Properties for Increments of $l^{\infty}$-Valued Gaussian Random Fields
This paper establishes general theorems which contain both moduli of continuity and large incremental results for $l^\infty$-valued Gaussian random fields indexed by a multidimensional parameter under explicit conditions.

Keywords:$l^\infty$-valued Gaussian random field, modulus of continuity, regularly varying function, large deviation probability
Categories:60F15, 60G15, 60G60

25. CJM 2008 (vol 60 pp. 457)

Teplyaev, Alexander
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18
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