The high temperature phase of the Sherrington-Kirkpatrick model of spin glasses is solved by the famous Thouless-Anderson-Palmer (TAP) system of equations. The only rigorous proof of the TAP equations, based on the cavity method, is due to Michel Talagrand. The basic premise of the cavity argument is that in the high temperature regime, certain objects known as `local fields' are approximately gaussian in the presence of a `cavity'. In this talk, I will show how to use the classical Stein's method from probability theory to discover that under the usual Gibbs measure with no cavity, the local fields are asymptotically distributed as asymmetric mixtures of pairs of gaussian random variables. An alternative (and seemingly more transparent) proof of the TAP equations automatically drops out of this new result, bypassing the cavity method.
Smoluchowski's equation is used to describe the coagulation-fragmentation
process macroscopically. Microscopically clusters of various sizes coalesce to
form larger clusters and fragment into smaller clusters. I formulate a conjecture about the nature of the fluctuations of the microscopic
clusters about the solutions to the Smoluchowski's equation. I also sketch the proof of the conjecture when the model is in equilibrium.
Abstract: (Thanks to work of Abel Klein and others) it is understood how
to represent the quantum Ising model in terms of a certain classical model
of stochastic geometry called the `continuum random-cluster model'. In the
regime of large external field, this geometrical model is subcritical. By
developing bounds for its `ratio weak' mixing rate, one obtains estimates
involving the reduced density matrix of the quantum Ising model. The
implications of these estimates for entanglement do not appear to be best
possible, but they are at least robust for disordered systems. [Joint work
with Tobias Osborne and Petra Scudo.]
I shall give an overview of reaction-diffusion fronts in
random flows, especially the variational formula of front speeds of
Kolmogorov-Petrovsky-Piskunov reactions. Large deviation of the random
flows is essential to the formula and the analysis of front
Consider a GI/GI/1 queue operating under shortest remaining processing time with preemption. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. Of particular interest is the waiting time for large jobs, which can be tracked using the frontier process, the largest service time of any job that has ever been in service. We propose a fluid model and present a functional limit theorem justifying it as an approximation of this system. The fluid model state descriptor is a measure valued function for which the left edge of the support is the fluid analog for the frontier process.
Under mild assumptions, we prove existence and uniqueness of fluid model solutions.
Furthermore, we are able to characterize the left edge of fluid model
solutions as the right continuous inverse of a simple functional of the initial condition,
arrival rate, and service time distribution. When applied to various examples, this
characterization reveals the dependence on service time distribution of the rate at which the
left edge of the fluid model increases.
The various concepts of volatility (realized, local, stochastic, implied), well defined or depending on a given model and/or statistical estimates, will be discussed. Backward and forward equations for call-option payoffs (Black-Scholes and Dupire equations) will be revisited. We will show that, besides the Black-Scholes model with constant volatility, fast mean reverting stochastic volatility models can reconcile local and implied volatilities. If time permits we will also look at the relation between volatility and correlation in the multidimensional case.
The talk is addressed to a general audience in Probability without any particular deep background in financial mathematics.
The totally asymmetric simple exclusion process (TASEP) is one of the
simplest models of interacting particle systems on the one-dimensional
lattice. It is equivalent to a random growth model from the
Kardar-Parisi-Zhang universality class. We focus on fluctuations of the
particle positions for a nonequilibrium TASEP that starts from certain
deterministic initial conditions. We (rigorously) derive the scaling
exponents 1/3 and 2/3, and identify the limit laws as those of Gaussian
Orthogonal and Unitary ensembles of the random matrix theory.
We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has power law tails. We show that for certain tail exponents, for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.
The ``random sampling'' in the title has nothing whatsoever to
do with statistics. Instead, it refers to the perfect reconstruction of
a band-limited function from samples, a classical problem of
Fourier analysis and signal processing. With deterministic sampling,
almost everything is known in one dimension and almost nothing
is known in higher dimensions. It turns out that one loses very little in
efficiency by using random sampling, and in return one can use
probabilistic techniques to get some interesting theoretical results.
This is joint work with Karlheinz Gr\"ochenig.