硬边缘 Pearcey 过程的间隙概率

具有可数状态集的硬核模型的吉布斯测度

Gibbs measures for a Hard-Core model with a countable set of states

本文研究任意阶凯莱树上硬核模型的非概率吉布斯测度，通过边界律方法给出平移不变与周期情形的唯一性和非唯一性条件。

Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential

We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order $k$ in the double scaling regime. Using the Riemann-Hilbert method, we obtain the large $s$ asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed…

硬边 tacnode 过程的大间隙渐近

Large gap asymptotics of the hard edge tacnode process

本文研究非相交路径系综中硬边 tacnode 过程的间隙概率，利用哈密顿量积分表示和微分恒等式推导大间隙渐近，并给出应用。

Asymptotics of spacing distributions at the hard edge for $β$-ensembles

In a previous work [J. Math. Phys. {\bf 35} (1994), 2539--2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large $s$ asymptotic form of the general $β> 0$ gap probability $E_β^{\rm hard}(0;(0,s);βa/2)$, provided both $βa /2 \in \mathbb Z_\ge 0$ and $2/β\in \mathbb Z^+$. It shown how the details of this method can be extended to remove the requirement that $2/β\in \mathbb Z^+$. Furthermore, a large deviation formula for the gap probability $E_β(n;(0,x);{\rm ME}_{β,N}(λ^{a β/2} e^{βN λ/2}))$ is deduced by writing it in terms of the charateristic f…

Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

Our interest is in the cumulative probabilities Pr(L(t) \le l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) \le…

Solving the Hard Problem of Bertrand's Paradox

Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article we show that Bertrand's paradox contains two different problems: an "easy" problem and a "hard" problem. The easy problem can be solved by formulating Bertrand's question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace's principle of insufficient reason i…

汉明空间中硬球模型的圈修正

Loop corrections for hard spheres in Hamming space

本文通过扩展置信传播方程和空腔边缘分布拟设研究汉明空间硬球系统的圈修正，证明最大堆积密度渐近达到Gilbert-Varshamov下界。

On the gap probability generating function at the spectrum edge in the case of orthogonal symmetry

The gap probability generating function has as its coefficients the probability of an interval containing exactly $k$ eigenvalues. For scaled random matrices with orthogonal symmetry, and the interval at the hard or soft spectrum edge, the gap probability generating functions have the special property that they can be evaluated in terms of Painlevé transcendents. The derivation of these results makes use of formulas for the same generating function in certain scaled, superimposed ensembles expressed in terms of its correlation functions. It is shown that by a judicious choice of the superimpos…