Fluctuations of the log-gamma polymer free energy with general parameters and slopes

@article{Barraquand2021FluctuationsOT,
  title={Fluctuations of the log-gamma polymer free energy with general parameters and slopes},
  author={Guillaume Barraquand and Ivan Corwin and Evgeni Dimitrov},
  journal={Probability Theory and Related Fields},
  year={2021}
}
We prove that the free energy of the log-gamma polymer between lattice points (1, 1) and (M,N) converges to the GUE Tracy-Widom distribution in the M1/3 scaling, provided that N/M remains bounded away from zero and infinity. We prove this result for the model with inverse gamma weights of any shape parameter θ > 0 and furthermore establish a moderate deviation estimate for the upper tail of the free energy in this case. Finally, we consider a non i.i.d. setting where the weights on finitely… 
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References

SHOWING 1-10 OF 47 REFERENCES
Scaling for a one-dimensional directed polymer with boundary conditions
We study a (1+1)-dimensional directed polymer in a random environment on the integer lattice with log-gamma distributed weights. Among directed polymers, this model is special in the same way as the
Tracy–Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder
The free-energy fluctuations of the discrete directed polymer in 1+1 dimensions is conjecturally in the Tracy-Widom universality class at all finite temperatures and in the intermediate disorder
Log-gamma directed polymer with fixed endpoints via the replica Bethe Ansatz
We study the model of a discrete directed polymer (DP) on a square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen (2012 Ann. Probab.
Borodin–Péché Fluctuations of the Free Energy in Directed Random Polymer Models
We consider two directed polymer models in the Kardar–Parisi–Zhang (KPZ) universality class: the O’Connell–Yor semi-discrete directed polymer with boundary sources and the continuum directed random
Stochastic growth in time-dependent environments.
TLDR
The Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance c(t) depending on time is studied and there is a transition at α=1/2, where the solution saturates at large times towards a nonuniversal limiting distribution.
Height Fluctuations for the Stationary KPZ Equation
We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data H(0,X)=B(X)$\mathcal {H}(0,X)=B(X)$, for B(X) a two-sided standard Brownian motion) and show that
Scaling Limits for Non-intersecting Polymers and Whittaker Measures
We study the partition functions associated with non-intersecting polymers in a random environment. By considering paths in series and in parallel, the partition functions carry natural notions of
Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension
We consider two models for directed polymers in space‐time independent random media (the O'Connell‐Yor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature
Tracy–Widom Fluctuations in 2D Random Schrödinger Operators
We construct a random Schrödinger operator on a subset of the hexagonal lattice and study its smallest positive eigenvalues. Using an asymptotic mapping, we relate them to the partition function of
Limit shape and fluctuations for exactly solvable inhomogeneous corner growth models
We study a class of corner growth models in which the weights are either all exponentially or all geometrically distributed. The parameter of the distribution at site $(i, j)$ is $a_i+b_j$ in the
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