Uniqueness of Gibbs fields with unbounded random interactions on unbounded degree graphs

Packing trees of unbounded degrees in random graphs

In this paper, we address the problem of packing large trees in Gn,p. In particular, we prove the following result. Suppose that T1, . . . , TN are n-vertex trees, each of which has maximum degree at most (np)/(log n). Then with high probability, one can find edge-disjoint copies of all the Ti in the random graph Gn,p, provided that p > (log n)/n and N 6 (1 − ε)np/2 for a positive constant ε. M...

Existence and Uniqueness of the Integrated Density of States for Schrödinger Operators with Magnetic Fields and Unbounded Random Potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinite-volu...

Pseudo-Random Function Generators With Unbounded Inputs

Definition: (See Goldreich, section 3.6.4.2 .) A function generator F with unbounded inputs associates with each n bit key k ∈ {0, 1} a function Fk : {0, 1}∗ → {0, 1}. We insist that Fk(x) be computable in time polynomial in the lengths of k and x. By pseudo-random for such a generator, we mean the obvious thing: the Distinguisher adversary D is given a function f : {0, 1}∗ → {0, 1} and can que...

Pseudo-Random Function Generators With Unbounded Inputs

Definition: (See Goldreich, section 3.6.4.2 .) A function generator F with unbounded inputs associates with each n bit key k ∈ {0, 1} a function Fk : {0, 1}∗ → {0, 1}. We insist that Fk(x) be computable in time polynomial in the lengths of k and x. By pseudo-random for such a generator, we mean the obvious thing: the Distinguisher adversary D is given a function f : {0, 1}∗ → {0, 1} and can que...

Uniqueness Theorems for Multidimensional inverse Problems with Unbounded Coefficients